Leak under bathroom sink at cold water shut off valve

[Cecilia Buckner]

I hate being in this position, but I, once again, am having some plumbing issues and I don't know enough about how everything works to figure out how to fix the issue.

On Friday, I noticed a small bit of water on the floor, in front of my bathroom vanity/sink,

and discovered some pooling in the cabinet - below what I think is the cold water shut off valve. My valves don't look like the ones I've found online... I tried tightening everything around the valve with a wrench but the water still drips. I'm not even sure to what extent the valves should be turned to the right or left to affect the flow of water from the faucet. The hot water valve seems somewhat tight to the right (not firmly, but somewhat) and hot water still flows from the faucet. When I turn the cold water valve all the way to the right I don't have cold water running from the faucet. Either way, with the cold water valve turned tight to the right or loose in the middle, it still has a slow drip. The water is coming from right beneath the edge of the white valve. There is some corrosion on the packing nut, but the nut is tight and the leak is coming from beneath the white valve. (Pics of the valve attached.) Does it sound like I need to replace the valve? If yes, is this something I can do without turning off the main water valve outside (I live in a condo and turning off the main valve would affect four other condo owners).

Think I'm going to look into some home repair classes after this . ...