I had what I thought was a similar problem. I was testing out my newly wired 56 and temporarily wired up my stop and tail lights to make sure my new harness worked. Tail and brake lights worked fine, but the turn signals would not. I changed out the flasher 3 times and it made no difference. I worked on this for hours and was getting frustrated. I was about to retrace all the wiring and create my own diagram so I could understand what was supposed to happen schematically. I am not sure why i thought of it, but I turned the key to power up the accessories and lo and behold - the turn signals worked fine. Short story: on my truck at least, the key must be "ON" for the turn signals to work. Brake lights and tail lights work whether or not the key is "ON"..
probably not your problem, but just in case....
