Question: All of a sudden we have a dripping bathroom fauce - more of a thin stream of water rather than a slow drip. But even when I turn off the quarter-turn Hot/Cold water valves located under sink, the water doesn't completely stop; it it just barely slows to a very fast drip-drip-drip-drip. What would be causing this to happen? Is it an issue with the quarter-turn valves themselves, which are only about 3 years old, or with the actual faucet fixture? Or could it be something else entirely? We have no access to main water shut-off within the apartment as it's a per floor/tier water system, accessible only by maintenance staff/plumbers in the building. Lastly, is there anything I can do in the interim to stop the water from the faucet until Monday when I can get a hold of maintenance? Thank you so much.
Sounds like you have a problem in both the faucet and the under-sink valves.
When the either the faucet is turned off, or the under-sink valves are turned off, the water should stop completely. If it still runs or drips when both are shut off, then both are not doing their job.
It is quite common for both the faucet and the under-sink valves to have rubber washers as the surface that seals when the valve is turned off. The most likely answer here is that the rubber washers have deteriorated or worn out, and are not making full contact to stop the water flow.
Alternatively, the valves may have a build-up of mineral deposits or rust, which is blocking the valve surfaces from making full contact to close.
The ultimate solution here is to fix both the faucet and the under-sink valves.
The fact that it started all-of-a-sudden sounds like something in the faucet broke. It is likely just a rubber washer that just needs to be replaced.
Shutting off the under-sink valves, reducing it to a drip, will relieve the pressure enough that you can open up the faucet to replace the washer. Depending on the make and model, there may be more than just a washer to replace; some brands have a valve "cartridge" that gets replaced.
Fixing the under-sink valves will require shutting off the water supply to the apartment. Not something you can do yourself, in your situation. You'll have to let building maintenance take care of that.
If you want to fix it before Monday, you'll need to shut off the under-sink valves, get your screw-driver or wrench, and open up the faucet to see if you need a rubber washer or something else, and replace it. Good luck with it!