Both calls and text messages use the cellular network, so this is almost certainly a carrier problem. Instead of going to AT&T call tech support. Carrier stores are staffed with sales droids, usually with little technical knowledge.
There are a couple of remote possibilities:
Do you have a 3rd party VPN installed, even if you are not using it? If so, delete the VPN profile in Settings/General/VPN & Device Management/VPN, then restart your device. Don't just turn it off; delete the profile. Updates sometimes "break" VPN, and it needs to be removed. If that fixes it, you can try reinstalling VPN, but probably shouldn't.
Is the phone in Airplane mode? Open the Control Center to check by dragging down from the battery icon.
Does the phone display SOS? If so, the phone is working, it just isn’t connecting to your particular carrier. If it does NOT display SOS it is a hardware problem, make a genius bar appointment at an Apple store.