headlights draining battery?

So I let my cousin take my truck to work because I taught her how to drive a manual with it and she loves it. The battery went dead on her twice while at work so I had to rescue her and my truck. It was deader than a doornail so I jumped it and let it run for a bit and it idled fine. So I turned on the headlights and headed for my night job. When I was almost there it was trying to die on me whenever I took my foot all the way off the gas. When I got to work I was finally able to get it to idle and I checked the battery. I'm getting 11.8 volts on this battery that I just picked up last month. Its a little low but still usable. So I turned the lights back on and watched the voltage dropping at a rate of .1 volts every few seconds and turns the lights off after the battery dropped down to 10.7 holy and the battery went back up to 11.8. It was raining and there is a piece of the running light lens missing on the passenger side. Is that the culprit or is the alternator dropping on me.

Sorry for the messed up words in the post. I'm using my phone and it likes to change them for me.

Bad alternator, sorry. Should be 14.2 Volt with a fully charged bat. If you check the voltage at the back of the alt. what is it?
But I think it's no good.

Well I have to say its been one of those weeks. I'll check it when I get home but Ill clean up the connections on the alternator first and if it does turn out to be bad Ill scrounge around pic n pull see if I cant find something that fits.

Cleaned up the alt and got the thing to run and took it to auto-zone to test it just to make sure. The voltage regulator was reading high and was only putting out about 12 volts. So I stole my wifes car and went to pic n pull. Found the jackpot there with a 97 Mitsubishi pickup that had hardly been touched. So I grabbed the alternator, wiper motor, a taillight lens that was busted on mine, the muffler since the heatsheild rattles on mine, and the speedo cable. The alternator I got works great. Puts out 14.2 volts. The truck seems to run a bit smoother.

Nice work. A low charging system does all sorts of unexpected things to a vehicle - wonder what the output amps were like...