Hi kristafrompotomac
welcome to the Apple (user-to-user) Community!
Re: "Facetime isn't working on my MacBook Air. How to fix? ... FaceTime is asking me to sign in using Apple ID
but when I enter my password the arrow just spins and spins and goes nowhere."
Check you are signed in (to your Mac) with your Apple ID: See: Sign in with your Apple ID - Apple Support
Then see if any of the several tips in Apple Support's online article (link below) can help:
If you can't sign in to your iMessage account or FaceTime on your Mac - Apple Support
Eg: says: "Make sure that the date, time and time zone are correct.
Set the date and time on your Mac"
_________________
Also, if needed: see: If FaceTime isn't working on your Mac - Apple Support
Eg: says: "Make sure you and your recipient have a fast Wi-Fi or mobile connection.
FaceTime calls over Wi-Fi require a broadband connection".
FaceTime Support website (info. / support / contact options etc):
FaceTime - Official Apple Support
All the best :-)