To Edit Mailing Address to use APO:

1. Open return (if not already there).
2. Click on Personal Info and edit the Mailing Address.
3. Under Type of mailing address in the dropdown choose Military (APO, FPO, or DPO).
4. Under APO/FPO/DPO in the dropdown choose APO.
5. Under State code in the dropdown choose your location.

for the state code on the dropdown select - 

Armed Forces the Americas (AA)

Armed Forces Europe, Canada, Africa, or the Middle East (AE)

Armed Forces Asia or the Pacific (AP)

Please see screenshots below.

We switched under those My Info to the APO address and it still will not allow us to efile.

It depends. Do you have an address for a very close family member you can use and trust? There is a question in the My Info section of Turbo Tax that asks where do you receive your mail and alludes to the fact that this may be different from where you actually live.