Water dripping from bathroom faucet even with the under-sink valves turned off?
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!