I think we'll need to know the Thermostat and perhaps "control box" model numbers as well as the rooftop unit itself.
Is it also a Heat Pump?
Personally, I'm not good at this, only mentioning what a tech or experienced DIY would need to know to try and help.
Our friend, member and tech Chris Bryant has a lot of manuals on his site.
Here's one on Duo-Therm troubleshooting but he has many more.
Clearly, it's OK, and common, to run Fan without Compressor. But the opposite is almost never OK. Many RV A/C units start both the Fan and the Compressor at once, together. That is the utmost amperage an A/C can draw, so sometimes the manufacturer starts Fan first so its starting amp surge is over before cranking up the compressor. It would be OK to start compressor first, too, but fan has to be running within seconds to keep system pressures under control.
If your fan and compressor are randomly switching on and off, that's a problem. If compressor's running without fan, that's a problem.
All I can offer in closing is this - Some wall thermostats have batteries, even though the A/C has control voltage to it. I know that in home A/C, bad batteries can cause the A/C to cycle On/Off.