To be honest, it sounds like your battery needs to be looked at first of all, because this looks just like the heart of the problem. Get it checked - NO, I don't mean just by checking the voltage across the two + / - terminals, I mean a proper load capacity test, which does as I stated, checks to see how well it will take in and hold a charge. It will show up immediately if there is a malfunction here. This honestly sounds like it came right out of the book of electrical diagnosis..
Al
Al