I am sure they know about their product, BUT what happens is that "people usually talk two different languages"... :-) they talk "technology / product" while you talk "user / product". Sometimes things don't "translate well".
You can likely follow their direction and get the AC model number from the actual outside unit (this is the big thing on the roof)... maybe a 5XX, or 6XX. However, the controller / relay board is usually inside for ease of access / repair. If you look at your picture of the plate, it indicates that there is wiring leaving the "box" and going up to the AC UNIT. There is also wiring coming from the thermostat and going into the "box".
Knowing what AC unit they installed will help as then when you look at the controller, it should be correlated as a "companion part".
IF you get a chance to talk with a technical person again at dometic, then the question to ask would be "what connections do the colors align with on the label?"... in your picture, you see the color codes of the wire and the connections saying "to thermostat"... each color aligns with some function going into the controller. As you saw in your picture of the thermostat, one can readily see a color to "function" for the wiring. What you need is that "equivalent" description from the label on the control box. IF those don't align, then you will having wiring "mismatch issues" which might be the source of your problem.
IT is strange that two AC units are operating the same way (ie. fan always on). It is unlikely that there is a component failure, more than likely a wiring error or possibly a relay card "mismatch" (unlikely but possible). From the dometic documentation, I am not able to determine the "color to function mapping". If you get this, it may go along way to solving your problem.
In terms of warranty, the AC unit (on the roof) and the controller (inside ceiling housing) are likely replaced together so indeed, it is a warranty issue if either fail. Your problem is the warranty probably doesn't cover "labor" which your "friends" don't appear to want to do for free (in terms of further troubleshooting). That's the "between the rock"... :-(
