Solved!
Didn't need to touch my other devices, aside from making sure they were all signed into the exact Apple ID and had the Caller ID pointing to the same number for incoming/outgoing calls.
Steps I took only on my MBA:
1) Close all running apps
2) Open system prefs app
3) Go to iCloud and sign out
4) Instead of keeping everything on my Mac, I decided to trust that the cloud had everything and delete each section of iCloud from my Mac (e.g. contacts, calendar, reminders, etc.)
5) Signed back into iCloud.
6) After reopening all the non-FaceTime apps connected to iCloud to make sure my data reappeared (it did), I reopened the FaceTime app and navigated to its preferences page where I selected "upgrade to wifi calling" at the bottom and followed the prompts to enter a code on my iPhone. Said ok to a few things on my MBA and it worked. The text inside the preferences screen mentioned it will now work even when my iPhone isn't nearby.
Hope this helps. It was a little bit of a leap of faith, but looks like signing out of iCloud only on my MBA and signing back in fixed it. I tested it by turning my iPhone off and was able to make and receive calls from all my devices.