Most leaking comes from the valve in the faucet, and most commonly mixing valves fail.
Mixing valves are used on one handle fauced setups, and mix hot with cold while controlling the amount of water coming out of the faucet.
First shut off all water to the faucet under the sink.
If your stuck screw is in the faucet handle, I would use liquid wrench there myself, and an impact driver if needed. If you have a single handle setup, once you pull the handle off you'll see the mixing valve which is usually held in by a horseshoe shaped pin with a little tab on the end wich prodrudes from the plumbing cylinder housing the valve.
Once that retaining pin is out you can ease the mixing valve out. My last one needed vice grips on the end to get out.
When you have it out, take it to Lowe's or you local pluming supply and get it replaced.
Good luck!