Why Dig? Trenchless Sewer Line Replacement Bypasses the Backhoe
First - NOT full septic tank - if that was the case you would be getting backup of sewage into the lowest drains in the house, and possible very slow flushing or refusal to drain out of the bowl - the opposite of your case.
Second - I assume you are the owner. If a renter or on a lease, this type of problem may be the responsibility of your landlord, depending on the terms of your lease or rental agreement.
OK - two possible situations here - low water in BOWL, or low water in TANK. I am assuming your toilet flushes OK, with adequate water to clean out the bowl, and that it is a typical type toilet with a tank sitting on the back of the bowl. If this is not the case and it is a designer toilet or looks like those at public restrooms (no tank), then the BOWL answers still apply if it is initially refilling OK, but if not enough refill water coming in at all then call a plumber.
First, low water in the toilet BOWL case. If the water in the toilet BOWL is low after flushing, I see four likely causes, in order of most likelihood -
1) the fill valve is not putting enough water into the toilet bowl. If you take the top off the tank, you will see a small hose (typically black plastic) coming from the fill valve (a vertical mechanism, usually at left side of tank, that the incoming water tube or flex hose connects to the bottom of on the bottom side of the tank). While the tank is refilling after a flush, a steady but not large flow of water flows through this fill tube and down into a vertical pipe or tube (usually brass or plastic and about 3/4 inch diameter, which stands almost full height of tank). The small tube puts water into this pipe, from where it flows into and refills the toilet bowl. This is also the overflow tube, which keeps the tank from overflowing if the fill valve fails to shut off. If the fill valve has a problem or the fill tube has a blockage, it may not be letting enough water into the bowl. Also, check the tube is actually pointed down into the overflow tube - if the clip came loose, rusted away or broke, then it may just be filling the toilet tank rather than the bowl. Check that a steady flow (will not be a real foreceful jet) of water is flowing out of this tube into the overflow pipe while the toilet tank is refilling. You should also see the bowl filling up at this time. If it come in but does not fill high enough because it does not run long enough, some fill valves have an adjustment - check fill valve manufacturer website for instructions. Others just have to be replaced - doable if you are handy at home repairs (see web videos on how to do it), or call a plumber for probably about $150-200 to replace fill valve (have him replace the flapper valve at same time if you get this done).
2) there is something like a rag or string caught in the trap (the waste passage within the toilet body itself) which is slowly wicking the bowl water down the drain - would be solved by a good snaking. If this is the case, the bowl will fill fully after flushing, but then slowly (typically many minutes to hours) drain down to just filling the start of the oval or round drain passageway where the waste passage starts to curve up into the toilet body.
3) blocked sewer vent pipe (which vents sewer gas and lets air into the sewer system so when you flush the traps in drains and toilets and such do not get sucked dry by the vacumn caused by the exiting flow. If this is the problem, then several drains in your house may have the same problem, or drain slowly. When you flush, the water will drain totally down the pipe and almost all the water in the bowl and trap will go down the drain too, typically with a gurgling sound for a few seconds at the end as the air seal is broken in the trap, then a small amount of water will flow back from the trap into the bowl, leaving you with water in the entrance curve to the trap but nowhere near normal height in the bowl - maybe not even enough to fill the entrance of the drain passage.
4) a crack in the toilet, letting water gradually leak out of the bowl onto the floor or into the subfloor. If this has been going on for long at all you should see water on the floor, or water coming out in the ceiling downstairs, or in the basement or crawl space under the toilet.
Case 2 - the problem is low water in the toilet TANK - since this is a sudden problem, two likely causes:
1) the float arm has corroded or the float setting has moved. Look in tank for any broken part. You may have a black ball on the end of a metal or plastic arm connected to the fill valve (which is the part, normally at the left side of the tank, that the flexible or copper tubing comes into at the bottom of the tank), or it may be a sliding cylindrical float that slides up and down on the fill valve (typically all plastic) - see if it is broken or loose or alll corroded up (for the arm type). When you flush, this float hangs down (if lever type) or slides down the fill valve (cylinder type), opening the fill valve so fresh water comes in to fill the tank and bowl. As the tank fills it lifts this float, till at the proper elevation the bouyancy of the float shuts off the fill valve. If the setting on this float has changed then it will either cause the toilet to "run" continually because it is trying to overfill the tank (float shuts off at too high a level, so water is continuously flowing down into the overflow tube and into the bowl); or it will shut off too soon, causing only a partial tank fill. There are adjustments to adjust the float shutoff setting - typically an adjustment xxxx on the arm-type, and a slider stop clip on a small rod for the sliding type. See web videos on how to adjust this, or call a plumber.
2) your flapper valve (in bottom of tank, the part a chain or cord or rod connects to the flush handle, which opens it when you flush the toilet, leaks. If it leaks AND the fill valve is working, the tank level drops till the fill valve opens, then the tank refills. This repeats at intervals, with the tank refilling periodically even though it has not been flushed. May need new flapper valve or just a good wiping of the sealing surface to remove grit that is causin it to leak. If this is the problem you will have a slight flow of water into the bowl continually, and will probably see a slight ripple in the toilet bowl.
3) water is leaking out of the fittings or bolt holes on the bottom of the tank. If this is happening enough to make you notice low takn water level, the tank will refill periodically the same as if the flapper valve is leaking, plus you will have water on the floor and dripping off the bottom of the tank.
Fill valve and flapper valves each cost around $15 if you do it yourself (you can buy just replacement flapper for less if that is the problem and the matching seal is good, but that is rarely the case). A plumber call to replace both probably $150-200, ASSUMING your water shutoff valve (at the wall, under the tank, with a flex or copper tube coming fromit up to the toilet tank) will work.. If it will not shut off the flow of water, then add another $50-150 to replace that, depending on how it is plumbed and whether he has to cut into the wall to replace it (rarely required). If you do go and have a plumber do it, have both the fill valve and flapper valve (and flush handle, if aluminum or brass and corroded) replaced at the same time, as all tend to go out with age - every 10 years or so. You don't want to have to call the plumber to replace another part in just a year or two.
From the sounds of it, you have a clog between the floor drain and the connection to the city sewer (unless you have a septic tank). The lower flow rates of sinks / showers / dishwashers probable don't cause a backup like the washing machine does. A couple of suggestions.
1. Snake the drain line with a spade tip snake, twisting the snake as you advance it. This should clear the partial blockage.
2. If feasable, have your washing machine discharge into a utility sink and put a strainer on the drain to catch the clothing fibre (fibres and grease from the sink probably made the clog in the first place not to mention a garbage disposal).
3. Replace your floor drain with one that has a backflow preventer (looks like there is a ping pong ball in it).
You have not said how old your house is, what normal water line life in your area is, whether it is leaking under the foundation or under the yard, etc. Cost depends a great deal on length of run, depth to dig to get below frost line, whether excavation will be through trees or other obstructions or open area, whether ground along route is too steep for a backhoe to work on, etc.
The first thing you really need to figure out is whether this is a spot repair issue, or a total line replacement issue. Your plumber should be able to help with that determination. Very general rule of thumb - very old line from before 80's, if galvanized pipe, could be at its practical life and be ready for total replacement. Copper line generally last about 50-70 years UNLESS in a corrosive soil environment or if it has groundwater flowing actively past it, then can be 20-30 years. Plastic lines from the 60's to 80's vary a lot - from as little as 10 years to 50 plus dependingon brand. Plastic lines (PVC, HDPE, PE) from the 80's and later are expected to last 50-100 years - most have not failed yet, so no good handle on how long they will last.
An above-ground or under the slab line a plumber will do. Buried line outside he generally subcontracts to an excavator to dig and backfill the line, or asks you to get the excavation done.