A trailer light tester is a nice thing to have. A voltmeter is pretty essential too, but the tester is better to verify the lights since the voltage jumps up and down. A tester is a great tool to have since converters themselves can go bad when hooked up to bad trailer wiring so it's nice to eliminate that first.
4-Way Flat Circuit Tester, Model# DBP | Light Testers| Northern Tool + Equipment
But at least one of the problems is on the trailer side. It sounds like at the least you have the turn signal and taillight wires mixed.