Q: Odometer/Clock turns on when car starts and then immediately shuts off. R/L Ac blend door actuators bad. Could bad actuator do it?

When key is in on position rather than start position odometer and clock both display however they cut off when car is fully started. I have replaced cam shaft sensors/battery/ignition switch. The only thing left to replace is the drivers side and passengers side heater blend door actuators. I have checked all of the fuses and they are all good. Could the bad actuators be causing the odometer to not display upon starting the car? I know that I replaced the evap solenoid as well. There is an electrical sound in the wiring harness under the hood near the battery when car is turned completely off. This would hint that there is still current flowing through the circuit even when the ignition switch is turned to the off position. Will replacing the actuators fix the problem? I know that the actuators still run even when car is turn off. They are jamming even after off switch has been pressed on a/c panel in center console. I also have cleaned all of the wiring connectors behind that a/c.

My car has 152000 miles.
My car has an automatic transmission.