starter problems???

I have a 1986 Caprice 305 V8. I went to start it and the starter stuck on and kept turning over even after I pulled the key from the ignition. It kept running til the battery died. I later tried to jump start it and couldn't get the battery to charge at all. Would the ignition cause this or would it be the starter? The alternator went out last year and I replaced that, and since have had to replace the battery a few times. What is killing my batteries?? Could there be a short somewhere?

Any help is greatly appreciated.

If You Had An Oil Leak And It Got Into Starter. It Will Cause Starter To Work Much Harder. Also Check All Grounds If You Are Eating Batteries Like That.