how do I find a phone that is shut off?

You can't track a shut off iPhone electronically. The only way you can find it is the old fashioned with your eyes way, so I'd not hold much hope you'll find your stolen iPhone. Thieves are smart enough to know a powered on phone, which is connected to a network can be tracked. So they turn them off and either pawn it or sell it on some site like offer up or Facebook Marketplace.

If you get a message your phone has been found and they ask you to provide your sign in credentials, DON'T DO IT. That's how they get you to provide the details needed to get in the phone and wipe it.

See this link --> Find your lost iPhone or iPad with Find My - Apple Support

Did you have Find My enabled on the phone?

Normally, the phone will not show up in Find My if......

The battery runs down

The phone is shut off

But.....on newer phones, the phone still sends out a very brief intermittent Bluetooth signal when it is off that might be picked up by another iPhone in close proximity that has an Internet connection. It's a really "iffy" deal, but works the same general way as AirTag.

So....it's possible that the phone might appear in Find My.....but it's also unlikely, to be honest.

No.