When the blower on the inside of the house cuts off, the air conditoner unit constantly runs.
The thermostat is new, so that is not the problem, the thermostat works with the heater. Any ideas?

Some switch has frozen in place. Until you can have it properly tested suggest you turn off the entire unit at the breaker at the main breaker box. This way you are not wasting electricity.

Sounds like you might have a stuck contactor,, but just because you have a new T stat don`t mean all is well with Tstat,