Is it possible to tell if a database has AUTOMATIC_TUNING switched on?
Automatic query tuning requires that Query Store be enabled first, then you can enable Automatic Tuning.

To check Query Store, select * from sys.database_query_store_options. You're looking for actual_state_desc to be READ_WRITE.

Then check the tuning options with select * from sys.database_automatic_tuning_options where name = 'FORCE_LAST_GOOD_PLAN'. You're looking for actual_state to be set to ON.