Problem fixed and knowledge learned by replacing the entire control system (Digital thermostat, control box including the board inside, and the coil temperature sensor.) The digital thermostat now uses fewer control wires between it and the AC unit. Since it is now a control signal instead of a control wire, it takes 1 to 2 seconds for the thermostat to tell the AC to switch fan speeds from high to low or low to high. The 1 to 2 second pause during fan speed switches is normal. What is not normal, which my unit was doing, is to run on low for a long time, then switch the fan off with the compressor running, then switch to high, then switch everything off. The new control equipment solved that problem: it now runs on low, then switches the compressor off, then switches the fan off a bit later. The other problem that I had, was that it would not stay running continuously on low (the fan would turn off at the end of the next compressor cycle)—that problem is also fixed—the fan will run continuously on low.
Since the unit was out of warranty, rather than fighting with Dometic about whether they had a defective board and should replace it free, I just went to National Refrigeration, Shipshewana, IN. Their prices are so reasonable it was well worth the cost for a no hassle fix. Leon is the best at AC and refrigeration.
Additional note: the thermostat is designed so when in auto it only runs on high fan when the air temperature is 4 to 5 degrees from set point. I have seen conditions where it runs continuously at low fan while staying 3 degrees away from set point. And, the only solution seems to be to lower the set point or force it into continuous high fan. Can’t say that I agree with this design.