Hey Maree130! I’ve been there, and maybe can help.
So are you saying your Mac doesn’t have enough storage to import the photos?
If the phone is so full, it’s shutting down, it may have to be erased using recovery mode:
https://support.apple.com/en-us/HT201263
Of course you’d want to save the data first.
If you do have an iCloud backup, and signed in on your Mac, you can check in System Preferences > Apple ID > iCloud > Manage iCloud Storage, you would see a backup there.
To check if your Mac has a backup of your iPhone, on the an earlier OS open iTunes > Preferences > Devices, you’d see them there, not sure in Catalina, but you can also access the hidden Library to be sure:
Open Finder, click Go on the top menu, if you hold Option on the keyboard, you’ll see “Library” about in the middle of the menu, click on that and open:
Application Support > Mobile Sync > Backups, if you see anything in there, it’s an iOS backup.
Anyhow, if all your photos are stored in iCloud, and NOT on your Mac, (And your Mac has enough storage to support them, (Or you can use an external hard drive, if your Mac doesn’t):
Quit Photos app, and re open while holding “Option” on the keyboard. Choose “Create New Library” and choose a location for it, e.g. External Drive or Internal Drive etc).
Once open, on the top click Photos > Preferences > General, and click “Use As System Library”, this way your iCloud Photos will automatically download to this Library,
In that same window, choose the iCloud tab, and put a check next to iCloud Photos, and to keep a hard copy choose “Download and keep originals).
Once downloaded, if you disable iCloud , and the Photos on your Mac and they are all still there, you can then restore your iPhone.
(Unless there’s other data you need to save first, as far as iCloud data is concerned, here’s how you can save most of that data locally to your Mac (or other device which I recommend):
https://support.apple.com/en-us/HT204055
Mac: Designate a System Library:
https://support.apple.com/en-us/HT204414
On your computer, you can see what exactly you has saved to iCloud by going to iCloud.com and signing in, (Note that Messages don’t show there, as well as some other data). That’s also a good place to go if you ever lost any data syncing to iCloud, there you can recover some lost data if it was saved and archived correctly:
https://support.apple.com/en-us/HT205018
Hope that helps! It can be a lot to look at, but if it’s the data you cherish most (As most of us do), there are many options, and resources to use, in the case your iPhone does need to be erased/restored, cheers!