Okay, tried a few combinations and still the same issues. After a little more investigation, I have determined the problem is between the ground and the green wire.
If I remove the ground from the Jeep, the light that stays on will shut off and none of the other lights will work.
If I go to the right trailer light and disconnect the green wires of the harness and right trailer light, then the light goes out, but the left trailer light still works. If I touch the green wire from the harness to the brown wire of the right light, the right light will light up (It's dimmer than with wires attached normally.) If I remove the ground from the right light the light is still receiving power because it is on (extremely dim, but on).
Also, with the 4 connectors connected and all my grounds in place but the jeep side, if I touch the white ground wire to the jeep frame it sparks. Similarly, if I have everything in place, but the ground coming off of the trailer 4-plug connector and touch it to the trailer frame, it sparks.
Is it a bad ground on the jeep side?
I'm going to contact the place I bought it from and see what's the deal.
Thanks for any other suggections.