All of the lighting circuits will have continuity to ground. There is resistance from the filament in the bulb, but you will still get an indication of ground continuity. An ohm meter would give you the actual resistance, but is not necessary for troubleshooting the wiring.
K Charles is right for the first step. Check the TV output at the plug.
I often use a spare battery to check the trailer circuits rather than hooking up to the TV. Since you may have a short I'd put a circuit breaker in the power circuit you set up to test with.
etrailer has plug wiring diagrams if you're unsure of the proper orientation.