Thanks to the encouragement and helpful suggestions from a reader I decided to give my rear turn signal another look.
I started with taking out the bulb and looking for any grime or corrosion that would prevent it from getting a good contact. Not seeing any I began to trace the line back to the tail light. Not seeing anything there I turned the blinker on looking for some sort of bad connection, and when moving the bundle of wire that go from the signal to the tail light it flashed, just once.
Well I knew where to start looking.
I started by taking apart the turn signal, thinking I was going to have to remove the whole assembly when I found my problem.
It took a lot of doing but eventually I got everything back together and working, the wire did not have much slack, and was a pain to reattach.
So I’m back to having a working right turn signal! (though I still had my integrated one the whole time hence my slight lack of pressing need to fix it)