To me it sounds like a duff battery that is going high impedance, or is going dry. It would probably be a good idea to check your battery fluid levels, and have a look at your charging system (fan belt tightness, alternator) to find out if it's all in order. A high impedance battery gives all the signs you are describing. This does not rule out the possibility you may have a short, however.

How old is the car battery? And out of interest, who did your installation wiring?