battery issues 1987 Mustang 5.0

ok I am going NUTS...my 87 5.0 has been having issues where everytime I go to start t he vehicle it will either slowly turn the engine and not start...or just make clicking sounds!!!

this is my second battery!! the first battery did the same thing....the second battery fixed the problem for about a month and then it started happening again..!

something has got to be draining my battery down!

would only an alarm system do such a thing??? I am really starting to get annoyed with this thing...

I'm pretty certain it's not the alternator because when I jump start the car and charge the battery....the battery will continue to start the car for about 12-15 hours after..!!