Batteries are draining

[BCinBC]

I just bought an automatic 1989 Toyota Corolla LE, a couple weeks ago.
I could not start the car last Thursday, but a jump start worked. Then the car would start during the day...no problem.
Friday morning I had to jump start it as well.
Same thing Saturday, so I put my battery from my truck in the Toyota and it Started on Sunday morning, no problem.
This morning(Monday), I had to jump start the car AGAIN!

Is this an alternator problem?
The lights do not go dim while driving.
The only thing is....the door ajar light stays on..all the time....even if the car is turned off.
The clock stays on too.
The radio has to be turned off when you turn the car off.
I am wondering if the radio has something to do with it?
Or can I just take the positive terminal off, while the car is running, to see if it is the alternator that is the problem?
The starter starts the car, without any problems, so it is not a "starter" problem.
What else could be the problem???

I checked the battery when it is running and it is at 13.50volts, so that is good.
I checked the battery about an hour ago and it was at 12.15 and now it is at 12.10volts.
One other question...how do I shut the "door ajar" light off?
Any help would be appreciated.
Thanks,
BCinBC