Is there a way to get an old phone backup that has photos I lost on a broken phone onto my computer or new phone (without replacing data on the phone currently)?

In order to restore data that is backed up to iCloud, you'd have to erase all content and settings from your phone. You would be starting all over again as if it were a new phone, and as one of the setup steps, would select the backup you wanted to restore.

To do that, you'd select settings > general > reset > erase all content and settings.

You might have a saving grace here: you could back up your current phone to iTunes and then erase it. Restore the cloud backup to the phone. Move the photos from the phone to the computer. Restore the phone as new. Restore the iTunes backup back to the phone. A lot of work, but if both backups use the same Apple ID, it would work.

If you have continued to backup to iCloud since the backup you want was completed, iCloud only maintains the last 3 backups.

Settings/iCloud and sign out or turn all syncing off.

System Preferences/iCloud and do the above.