They work fine when the headlights are off, but when the headlights are on, the tail lights on the trailer turn off when I hit the brakes. They do blink on left and right turn signals. Is this a problem with the tail lights?

sounds like a bad ground wire to me. Check for a good ground source on the trailer and at the plug.

I agree with skoryaro2, if it is a new truck probably a bad ground on the trailer or perhaps trailer not wired to code.

A lot of older trailers didn't actually use a ground wire in the light connector, they just grounded through the tongue, doens't work! I'm just about certain that is your problem. Run a #12 ground wire (white) from your trailer plug to the frame and use a good star washer and bolt to attach it with. If that doesn't fix it, then make sure all your lights are grounded to the frame.

Check your truck with a different trailer and your trailer with a different truck. Then troubleshoot the one that doesn't work.

I have a similar problem. With the headlights off, everything works ok.
With the lights on, everything works, except when I use the turn signals.
With either right or left turn signal on, the trailer lights flash right and left alternately. Hazzards work, brake lights work, tail lights work. Weird!