Startup Disk is full but can't do Dropbox
Are you syncing your Photos app using iCloud? Using that in conjunction with the setting "Optimize Mac Storage" should solve your problem (provided you have enough iCloud storage space). When set up this way, Photos stores your entire library on Apple's servers, and only downloads full-resolution photos and videos onto your local hard drive as you view them. If you have a lot of empty space, it'll cache more of them locally, but if you start to run out, it will get purge the high-res versions to make room. The thumbnails remain for browsing. 

The big advantage to this over moving things manually onto Dropbox, external drives, etc etc is that your photos and videos stay right there in Photos, and are available on all your Apple devices without any real work on your end. Everything is backed up, and you aren't spending all this time manually shuffling things around. The one (and only, IMO) disadvantage is that you need to pay for whatever level of iCloud storage suits your Photos library size.