I disconnected my trailer relay and measured the voltages, drops, and tested it along the signal/brake light circuit, both while it was operational and while it was apart. (This was not a big a job, it took about ten minutes.) I also tested it using a really simple 12v test light, since it's almost impossible to hurt anything on a car's circuit using a test light -- so that's what I'm going to recommend you start with.
So, your first step to fix the circuit is to carefully cut back the insulation from the green and yellow wires that come off of the "trailer side" of the trailer relay (not the "car side" of the trailer relay that also has the red wire going in to it). Do not cut the wires in two, just shave the insulation off of them. Then, with those two wires exposed, attach your 12v test light to a good ground and then to each wire, one at a time. Test your brakes and turn signals with the light attached to each wire. See if the test light behaves correctly or if it misbehaves like your car's lights do. Put it through all the paces by testing it with other lights, like your headlights, on as well. Take notes and report back.
There's an unusual aspect of the trailer relay that you should be aware of: There is a small amount of voltage that "bleeds" over from one turn signal output to the other. A volt meter will picks it up and your test light (mine did) might pick it up as well. It's not enough to fire off your car's lights though. What this means is that if you have your test light on the yellow wire, and the turn signal is flashing the green wire, you may see a very dim flash on your test light when it's attached to the yellow light.
This is normal.
Digital volt meters are not the best for dealing with flashing circuits. An old fashioned meter with a dial and a needle is better, and a test light is best.