Have you put a set of gauges on the system to read the actual pressures? There is a low-pressure switch that will drop out the compressor below around 22-28psi.
You didn't mention the year, but some of the newer vehicles with R134 freon also use a high-pressure cutout switch.
Are your condenser (in front of the radiator) and evaporator (inside the HVAC fan / heater enclosure) coils clear?
Not sure if your model year uses a cycling clutch or an expansion valve setup. It is normal for a cycling clutch system to kick in and drop out if the cooling load isn't too great - however, it sounds like this isn't the case. Also, may have a blockage somewhere in the lines.
If you haven't worked on A/C systems before, my advice would be to take it to a professional. The extreme temperatures and high pressures involved with AC systems can cause further equipment damage and significant personal injury if the work is not done properly.
Good luck, and keep us posted on your findings.