Your problems may not be related.
Check the condition of your battery first for the current drain problem. It may be FUBAR.
As for the reverse light, remove the block connecter in the boot from the light clusters and look at the terminals. If you see one very corroded, then the voltage will try and earth itself through the next easiest route. That is why sometimes when you are stuck behind an older car and they indicate, you get a mini light show. Same if they reverse or brake.
Clean the corroded terminal and plop a blob of solder on the corrosion damage. Blend off and refit.
