For one reason or another I seem to be overly biased towards my own recommendations

so I will emphasize what\why I suggest you follow my recommendations prior to purchasing a new A/C for your RV.
Many RV air conditioners do not have a "Freeze Protection" sensor (especially older units) and as long as the air flow is good and the air not too humid, no icing problems would occur. However, A/C's still froze up and certain installations (ducted for example) caused the icing problem to become even worse so the solution from the A/C manufactures was a simple solution which was to install a temperature probe into the "cold" (evaporator) coils and automatically turn off the compressor (leaving the blower fan on) until the ice has melted away (which increases the air flow as well) and then turn back on the compressor. What you should note when it is working correctly is that the blower fan continues to run and the compressor will cycle on and off, turning itself off whenever icing is detected (even if the RV interior has not reached the thermostats set point). (btw, A/C fan speed should be on HIGH especially during icing conditions (high humidity locations for sure) and the fan should be set to "ON" instead of "AUTO").
In any case, while there are certainly reasons such as low air flow and low refrigerant that can cause the coils to freeze, the point many are missing here is even if low refrigerant is the root cause of the icing (which I highly suspect is not the case here), none of those reasons account for the fact that icing should still not occur with this unit because as any icing occurs the A/C compressor (which makes the cold) will turn off prior to icing over the coils very much at all "IF" the freeze protect thermistor and control is actually working.
Recommendations:
I highly recommend replacing the "Freeze Control Sensor" as it is a common part and when it goes out the problem you are having is common. Also it is much less costly at only $10 and night and day easier to install than a new A/C. I will also add that any A/C that has a freeze control circuit, and the A/C is freezing up, then I would suggest that the failure is with the freeze control system that is supposed to prevent the freezing (regardless of any contributing causes to the freezing). ~CA
I have very high confidence this the correct replacement part you need and I also have high but a little lower confidence that this will fully resolve the issue as you could have other issues as well, although other issues would be less likely. I would still take the time to protect the exposed wire, clean the coils (both), and inspect everything visually as well as possible. (also to add, you can test the sensor with an electrical meter which I would commonly suggest but I don't know if you have an ohm meter or know how to test it and in addition, simply replacing sensor is likely needed anyway and again can be done relatively easily and for a very low cost.