Al, before anything else, check the connector and make sure it is not damaged and the wires are intact. I would speculate that you have a bad ground. Most trailers are set up so that the trailer frame acts as a large part of the grounding circuit. Typically, the white wire that comes from the plug on the trailer goes back a short distance and then connects directly to the trailer. At each light you will have a short wire that is also connected to the trailer. Often times, the lights will work even though the ground is insufficient, because it will ground through the hitch ball. As the ball rusts (or the trailer), you may lose the ground. If you do not have a 12 volt test light, go buy one. Try to find one that has a sharp probe in the handle. Us the sharp probe to penetrate the insulation on the brown wire of the vehicle harness. Insert the other probe into the connector socket of the white wire. Your test light should light if the vehicle running light circuit is operable. Turn on the left turn signal and penetrate the yellow wire. The test light should flash with the turn signal. Do the same for the right turn signal/green wire. If all checks out good, connect the trailer plug to the vehicle. Do not hitch it up. Connect the test light to a sufficient ground on the truck, such as the hitch or the frame. Penetrate the insulation on the brown wire (trailer side of connector). If the light comes on, move the test light ground to a good ground on the trailer (typically a screw that is screwed into the trailer frame or a bolt that is digging into the paint enough to touch metal). If the test light comes on, then the connection is good. Take the test light to any of the running lights and connect the light ground to the trailer.. Use the sharp probe to penetrate the brown wire. If the test light comes on, the wiring and ground is good. If not, then check all of the grounds and the wire between the tongue and the light you are testing......
Man, I really complicated this response.....