Is the original or old thermostat really compatible with the new air conditioner?
Also, most newer air conditioners run in such a way that once they shut off, there is a delay before the cooling compressor will turn on again. Depending on the air conditioner, it could be as long as a 2 minute wait. But, during that wait time, the fan usually does not turn on either.
It sounds to me like there is a miss reading coming from the wall thermostat. If it were mine, I'd start troubleshooting there. The air conditioner gets its command signals to turn on, turn off, run on fan, reach it's temperature and turn off, all from the thermostat. The air conditioner, if it were just plugged in and wired with no thermostat, it would just run and run and run. It's the thermostat that controls on-off commands.