On some of the photos, a message says “an Internet Connection is required to download larger versions for printing.”
You will be seeing this message a lot, if iCloud Photos is enabled in the Photos > Preferences > iCloud, and also "Optimise Storage" and if you are having more photos in iCloud than will fit onto your Mac. Can you confirm, that iCloud Photos is enabled in the Photos Preferences? And "Optimise Mac Storage" then?
Then compare the amount of Photos you are having in iCloud to the storage capacity of your Mac.
Do you still have plenty of "Free" storage, at least 20GB (you can see this, when you launch Disk Utility)? The Finder will only show you "Available" storage. That does not tell you, how much storage is free, because it includes the storage that can be freed by purging photos from your library.