Had my old Mac Air wiped clean and now I can’t login.

If you wiped the drive clean, there is no OS. If you can attempt to login but it won't accept your password, the drive either wasn't wiped or you've just forgotten your password. A login screen can only be reached once an operating system has been installed, and after wiping a drive that would be the setup screen, not your normal login.

What exactly did you have done to your drive? Being "out of memory" is not the same as being out of storage ... disk drive space.

Wiping your drive is not necessarily going to give you more space if you restore from backup ... you just put what was filling up the drive right back on it. If you don't restore from backup, you have more space, but you'll start taking that up as soon as you start using you Mac.

It would have been far better to add external storage, HDD or SSD, rather than wipe your drive.

If you have a login screen now, you have a working OS. At least at this point, and that means you just forgot your password.

Go here for Apple's definitive solution for forgotten passwords: https://support.apple.com/en-us/HT202860