Where is your Photos Library located on your new MacAir-- in the Pictures folder? How big is it--mine is 130GB. Is there still a copy of the Photos Library on the old MacBook Pro? You need to be sure that the Library there is not harmed!
How much free storage do you have? Check System Settings>General>Storage. You need at least 10% of your disk storage to remain free.
Are all your pictures in iCloud Photos at iCloud.com? Be sure to look and make sure.
Your best bet may be to trash the Library that's on your MacAir, and make a new empty Library. Make this your System Library, connect it to iCloud, and then sit back and wait for it to fill with all the pictures from iCloud. You should probably answer all these questions for us before you start that, though.
Let us know…