My AirPods don’t work with Apple Watch on, it only works when I turn off the watch

Go to Settings/Bluetooth. How many items show as Connected? Bluetooth has a limit of 7 simultaneously connected devices. If you already have 7 (or 6 if you are trying to connect AirPods, which use 2 slots) you need to disconnect one. You already discovered that by disconnecting your Watch. If you have the Tile app that can use a lot of slots, however, there is a setting in the Tile app that will limit the connection to one device at a time.

I had the same problem. Removed all Bluetooth devices except for the Apple watch and Airpods. Only if I switch off Bluetooth on the watch will the Airpods work.