That is screaming wiring harness problem. Or keyed ignition not getting power to run post.
Lights normally work without key. But signals not normally unless wired off of battery hot vs ignition run.
if your are running a relay for the starter and its not getting run hot (vs start key position hot) it would explain that.

Fuses etc.
I would just start with the volt meter working your way from battery down the run hot paths first and see when it stop producing power.
And look for any unhooked wires.