Battery becoming very hot

Check all the running insert-and-twist bulbs to make sure they are the correct bulbs in the correct sockets, and correctly installed. If you don't find an issue there, check the side markers. 194 type bulbs have been known to short if not carefully installed.
On bulbs with two filaments, check that they both light independently, at proper intensity; I have seen those shorted inside the envelope, one to the other, or a pool of melted metal in the bottom.
If the needle bounces around with the brake lights lit, check those first.
If everything turns out hunky-dory, start pulling fuses, the problem might be in the signal switch itself, which is a type of signal router. The cheap aftermarket ones are known to melt, and anything could happen.