Thanks for all the suggestions! I bought a test lamp and tested at the flat 4 wire and all lights worked properly on the test lamp.
I did some more testing with the ohm meter, and found that with all the lights turned off, on the flat 4 connector coming out of the CURT wiring harness that both the white and the brown(tail lights) wires have zero resistence with the car ground. Would this mean that the CURT wiring harness is probably bad? I still don't understand why it is working with the test lamp, and why the blinkers work on the trailer if I hook up the Reese adapter ground after the blinker is on.