I didn't know you had a landlord...
Send him a letter, let him fix everything. Tell him how much you love this place, and that you want to stay here forever (true statements right?). The point is clear: put it in writing.
can't find the make on the faucet. looks like an old american standard. landlord didn't go in for expenses. stainless steel. hot and cold water knobs are that plastic cut crystal look like you see in bathrooms. the sprayer is white. all i know is it's getting worse and i don't want to even hassle with the landlord. i just want to find out if it's something i can fix and take care of it. any help would be greatly appreciated.
Is this the faucet you have?
You don't want to get the landlord involved. OK, you're a "good" tenant. But...
John is giving you all the free help you can ever get, and I suspect that you're still unable to fix your problem. More than a month after you first posted, your drip is getting worse. Replace your faucet.
Your no name faucet is very inexpensive. You can find an as like replacement for less than $30. If you to want to upgrade to a Delta, with lifetime warranty on parts, get a Peerless (a Delta compay) at Walmart for $35. DONE.
And always remember, if the job is over your DIY skills, hire a handyman.