A quick test to determine whether you're dealing with a short or other unknown drain on the battery:

With the car off, disconnect the negative battery terminal. Connect a test light between the [removed] terminal and the negative battery post.

If the light comes on, there's an electrical draw. If you have a clock, or a stereo that uses a memory wire, or any other thing you know of that's always active, disconnect it. If the light stays on, you have a drain. Remove the fuses one at a time - if the light goes out when you remove one of the fuses, you can then narrow your search to include only the circuits connected to that fuse.

This won't do anything to fix a dud battery, nor does it address whether or not your charging system is or isn't adequate. But it'll allow you to find out whether any of your symptoms are related to a constant drain on the battery.

Best,

Cameron
Rose City