I want to redeem the $4.99 monthly credit offered in email but the verification code took 15 minutes to arrive. Do I need to log out of Apple TV to activate the credit?
See above. Perhaps that's the subject line? Oops
On what device is this happening?
You need to be signed in to redeem the offer.
You should redeem the offer on your new Apple device, not a different device. Once you have the subscription, you can use it on any of your devices with the Apple TV app.