Turn signal with lights on

Are the tail lights LED?

One lamp on the affected side is not grounding properly. I have had this happen 3 times

Twice it was the front turn signal lamp on the motor home. I removed and replaced the bulb and that was it.. THe lamnp itself was not connecting properly to the socket

The 3rd time was after instaling tow lights on the towed car.. Turns out I needed to imnprove the GROUND at the motor home's socket

If it was only the indicator lamps, I would say a bad lamp. With the other issues involved I would suspect a bad ground.

It sounds like a bad ground at the head light. With no ground at the head light it lets power go to the ground side of the signal light. When the signal light tries to come on it now has voltage on both sides so no blink.

Many odd situations in automotive lighting arise from bad GROUND connections. When a lamp is NOT powered ON, it may serve as a ground for another lamp that has power applied but whose ground has failed. This causes odd results.

I would look for a heavy duty flasher

I would always start with the lens covers off the back bulbs so you can see the filaments.

While parked, look at the filaments and see what comes on when you try and turn. If you have a double filament bulb, the thick filament should come on and off.

If you are seeing the thin filament coming on and off, then try grounding the side of that bulb. If that fixes it, then fix the grounding where it is loose.

If that isn't the problem and you have taken off the red lenses and checked, come back and write some more.