Ac only works when the car is not moving and the gas engine is not running

Did you use a Toyota Techstream diagnostic system or follow the procedure in the Repair Manual (more info) to check for codes using the air conditioning control panel? If you used a generic OBD II scan tool designed for engine and emissions diagnosis, it probably wouldn’t have been able to communicate with the air conditioning amplifier assembly to retrieve stored codes.

In any case, @Leadfoot J. McCoalroller's suggestion above about the air mix damper (the No. 1 air conditioning radiator damper servo sub-assembly, in Toyota’s terms) is a good one. If this damper is broken or in the wrong position, the airflow through the air conditioner will go through the heater radiator unit sub-assembly, which has hot coolant when the engine is running.

Even if the code isn’t stored, I’d suggest the Repair Manual troubleshooting procedure for B1441/41, Air Mix Damper Control Servo Motor Circuit (Passenger Side). If it is the servo sub-assembly, though, replacement is unlikely to be a do-it-yourself job.