A 97 Dometic would be the older Brisk Air.
If your T stat has a fan "only" mode, try the fan-only option; no need to unhook wires in this case. As was said, if the fan then runs, the compressor can have issues.
If the compressor is toast, it's time for a whole new unit. The compressor gas system really is not made to be serviced unless you are a trained HVAC tech. And even then, there are gas system parts issues trying to get them.
And here is a heads-up on an entirely new unit. Dometic changed the design of the Brisk Air to the Brisk Air II. And in doing so, while the new Brisk Air II will fit in the same 14" x 14" hole in the roof, the air box in the ceiling, the control box, and the t-stat also have to be changed if you have a ducted system. If you have a non-ducted system, the air box in the ceiling still needs to be changed. They did not make the two generations of air path from the roof unit to the air box compatible.
After seeing the new Brisk Air II redesign, I am not super impressed with the longevity of the styrofoam system. I changed to the Coleman Mach brand on the last AC system I replaced. As of now, in my view, they are built better with a better warranty. You still have to change the air box and controls to convert to the Mach unit, but you are doing that on the Dometic.
Hope this helps
John
2005 Ford F350 Super Duty, 4x4; 6.8L V10 with 4.10 RA, 21,000 GCWR, 11,000 GVWR, upgraded 2 1/2" Towbeast Receiver. Hitched with a 1,700# Reese HP WD, HP Dual Cam to a 2004 Sunline Solaris T310R travel trailer.