I am having the same issue with my 2006 Sierra. The fuses under the hood were fine. The right turn fuse (panel located lower driver's side of dash) was burned out. I replaced it and that did not help. I have read that the flasher unit on GM trucks is prone to failure. I suspect that this may be the problem. An after market flasher will be cheaper than the OEM, and reportedly more reliable.
The wiring from the front to the trailer connector is all good and grounding is also good. Have not yet replaced the flasher unit, so can't be positive about the outcome.
Hope this was of some help.

Check the connecter cable

From your description it would seem the towing electrics have been incorrectly wired or perhaps the problem is with your trailer.

My advice is to go back to basics, get yourself a diagram and investigate as deep as necessary. Your car is probably filled with digital electronics including a multiplex lighting system. If the towing electrics are an official GM kit there should be no problems but if it has been wired the old fashioned way a multiway relay should be used to prevent the trailer drawing/interfering with the multiplexed vehicle lights.
That is the safest way but it does mean the circuit becomes more complex.