iMessages stuck at sign-in on MacBook Air M1

Not certain, however you may try force-quit, and then restart into startup

with Safe mode; that 'may catch it' off-guard. [But it may not.] Commands

to do things, are different with Apple silicon and its newer M1 series S.O.C.

• Start up your computer in safe mode [M1]

1. Choose Apple menu > Shut Down.
2. Press and hold the power button on your Mac until you see “Loading startup options.”
3. Select a volume.
4. Press and hold the Shift key, then click Continue in safe mode.
5. The computer restarts automatically.

• How to use Safe mode on your Mac - Apple Support

https://support.apple.com/en-us/HT201262

There are several newer different things listed under macOS Recovery

for M1 Silicon ~ some/all appear to have different keyboard shortcuts.

Instructions in above linked site, start out similar to those of Safe mode.

[Then each those, go a different turn. You may have to read & experiment.]

• Use macOS Recovery on a Mac with Apple silicon - Apple Support

https://support.apple.com/guide/mac-help/macos-recovery-a-mac-apple-silicon-mchl82829c17/11.0/mac/11.0

(Each of these with links, have been 'manually copy-pasted' as separate parts.)

Certainly there likely are additional resources. Several older pages list 'Intel

based Mac' content. Some have both answers, in one: for M1 & Intel-mac.

• How to Boot an M1 Mac into Recovery Mode (Compared to an Intel Mac)

https://eshop.macsales.com/blog/74502-boot-an-m1-mac-into-recovery-mode/

• How to Force Restart M1 Mac - OS X Daily

https://osxdaily.com/2021/02/08/how-force-restart-m1-mac/

..Kind of harder to guess (unless on luck) you find the trove of M1 gems..

Anyway, my reply may or not be good; some are off by quite a margin!

Take care & safe travels! 🌤☺︎

Glad to hear of your success!

I've found odd issues w/ use of iCloud settings.

Some oddities due to way different iDevices

and Macs; plus different iOS/macOS versions.

Take care! ☺︎