Answering this requires Model Numbers for both A/C units and some research. Or Models and some wise input from an experienced tech. Of whom we have at least two, Chris and Doug. It may or may not be simple to switch the control system from one A/C to the other.
Nice thing about Coleman is access to parts lists, wiring diagrams, etc. Dometic may be the same, and I just haven't had a need to look.
Three basic things can go wrong with RV A/C's
1. Compressor may fail. This is what we all fear and it's relatively rare.
2. Sealed System (coils and tubing) may leak the refrigerant. Also rare.
3. Something Electrical
Replacement is appropriate in 1. and 2. since most of us can't do A/C repairs and an RV Tech will cost as much as a new A/C for DIY install.
But 3. can be something very simple and inexpensive. Thermostat, Switch, Wiring, or the VERY COMMON problem, failed Capacitors. Your Compressor depends on a Run Capacitor. It may also have a Start Capacitor. If it has Start, it'll also have some form of device to drop Start out of the circuit once the Compressor starts.
We can help with troubleshooting the one that is "isn't blowing cold" but we'd need more symptoms and you to end up on the roof with shrouding removed for inspection and some simple checks. You'd need a Spotter to open the Breaker for part of the checking and close it to see if you made a difference.
Coleman fan myself. We had one cool (and remove humidity) in FL over 25 years. Our current one does the same, just not as old yet. Dometic may well be better and I'm just ignorant. That said, Parts and Service are very important to me. I'll settle for a pinch less performance if the service support is better.
OH... This is a better Tech Issues question than a Class C question. A/C and related problems are common to all RV types, and the techs tend to cruise the Tech Issues forum more frequently. You can Contact Moderator and ask the thread to be moved...