13.1 is too low unless the battery is serious under charged and the charging system is trying to bring it up.

Which is why I said to measure the battery without the engine running first.

I would put a charger on the battery and let it charge up fully, then run the voltage checks and see where you are.

As for your directionals, assuming you are using a stock type flasher it requires a decent amount of current flow in order for the flasher to start flashing, low voltage will stop it from working.