Apparently the signals and hazards get different power when the key is in the park position, I would assume there is a different fuse protecting it.
It is out of warranty by 6 months.
As for the relay, since it seems to work with the key in park it seems unlikely to me that it is the relay, but if anyone can give me instructions on testing it I'd be happy to try. It's a $70 part so I don't want to just buy another unless I know I need it.
I have disassembled all turn light sockets and replaced all bulbs and the fuse blew on the way home from the parts store where I put the bulbs in. The light sockets looked to be in good condition.