Turning cellular off only turns off cellular data, ie Internet and iMessage. So you won't have data usage charges. It does not turn off the ability to make or receive phone calls. That's why you are seeing the ATT. However, if you don't make or receive a call there are no charges with data turned off.
Airplane mode will work but it also turns off wifi and you would have to manually turn on wifi. I'd just leave things as they are with cellular data off and don't answer the phone. That's what both my wife and I have done for the last number of years and never a charge.