Your photos are not of the "Relay Module" (known as the Relay Control Module), what you have there is the "Pulse Module".
Going to dealers is a waste of time, stop doing that. (See my sigline, obscenities starting in 3, 2, 1..............)

You can send the Pulse Module to D.J. in Arizona, he can replace the bad relay for around $40 including shipping back to you.
I'd go that route, myself. I really would.