Lets take your first post
1. There is no way your AC unit can start up and IMMEDIATELY blow cold air. Most RV AC units have a delay of 15 to 150 seconds before the Compressor is powered up.
2. What you feel is the compressor locking up IS THE COMPRESSOR starting up. Either the Start capacitor is bad or the compressor is bad. On a 7 year old Dometic. I would suspect the compressor is bad. IF it was a RVP, then the Capacitor would be suspect.
3. The thing to do is install a new Start Capacitor. IF the Compressor starts and runs, then the capacitor was bad. If the Compressor amps out(locks out) with a new capacitor, the compressor is bad. Doug