The engine cannot turn over if the battery is flat, as this is what powers it!!
If you are having electrical problems yet your engine turns over it is not the battery causing them.
Fully charged and running wants to be a little at around 13.8-14.2 volts no load at idle.
Next with it still idling put al lights etc on and especially HRW and then raise revs to 2000rpm then it should read at least 12.8 or very close.
The diodes could be starting to leak in the rectifier pack, these stop AC signal from getting to the battery which the alternator produces as well as of course converting the feed to DC for battery.
When these leak they allow current flow from battery to alternator, ie the other way around.
It could be a duff battery causing it try charging the battery then leavingit disconnected for a day, then reconnect and see if it is ok.
A simple and cheap way to check for current flow with everythingoff is to use a small bulb temporarily between a battery terminal and it's lead, any current flow will illuminate the bulb, anything more than a faint drizzle is not right, make sure doors are shut though!!
tabetha