Concerning the air recycling. I did look into that and the unit did have some small openings (3/16" x 3/4") at each corner between the suction and discharge. The weather striping in other areas appeared to be in good shape. I closed off the leak areas with metal AC tape. However the quick restart after shutting down makes me suspicious of the thermostat. It's not easy to determine the actual deadband that the thermostat has. But, I'm going to try an optical thermometer.
Also thinking about trying the unit without the ceiling shroud and bending the sensing bulb to be outside the airflow area. I don't want to ruin the thermostat.
I finally found a time delay relay that I think will work. Now to go to Corpus and see if Altex has one.