apple watch stopped unlocking MacBook air

See this support document for the steps to activate that feature. Unlock your Mac with your Apple Watch - Apple Support

I understand that you said it worked before, however during updates, things can get messed up for lack of a better word. Go into System Preferences, Security & Privacy. Uncheck the box to use the Apple Watch to unlock the Mac. Restart the Mac. Then go back into the same place and turn it back on again. After restarting, you will have to log in with your password, but after that, the Watch should unlock the Mac. I have to do that every once in a while with my 2019 MBA and my Series 6 Watch.

That I cannot assist you with. For that problems, I suggest that you start a thread in the Big Sur forum for assistance with that issue. Generally, removing the checkmark in Security and Privacy, then restarting the MBA and checking it again will work. I've no issues with my MBA 2019 with Big Sur and my Apple Watch Series 6.