Here, I did a five minute cut and paste on your car's wiring diagram so you can see what you are dealing with. Since your front flashers work, you know that your signal wand switch, and the flasher unit, are both probably working properly. Your brake lights and signal lights use the same filament. So, from that, you can surmise that:
1) If your brake lights are working normally, then you know the connections from the trailer relay to the rear bulbs are good, including the grounds.
Lights OFF - both brake lights work;
Lights ON - both brake lights don't work (along with the flashers on both sides), although the tail lights do work.
2) If your brake lights are also misbehaving, then it could be either the trailer relay, or the connections between the relay and the bulbs, that includes the grounds.
Therefore, based on response to item 1 - it appears to be the 'trailer relay' or the connection between the relay and the bulbs, including grounds!
You will need to test the voltage, using a VOM or test light, coming out of the trailer relay, on both the yellow and green wires. If those readings are good, then you will test the wires as they go in to the bulb fixture itself (note that it appears the wire colors change). If we have to, we'll also check the wires going in to the trailer relay and maybe use a 12v jumper wire to test the bulb circuit itself. The way you will check the wires is to carefully scrape a little bit of the insulation off of them, using a razor, so that you can see the copper conductor. This is really not hard at all.
If I had to guess, I think it's going to end up being either a bad trailer relay or a bad connection (that includes a ground).
