Using an USB external floppy drive, I've been able to use my Mac to back up files from a big box of 3.5 inch floppies (800K, HD 1.4MB). But I cannot even read my DS 1MB floppies using either the external USB drive, or the drive of a Classic II. (The Classic II kept freezing up). What kind of drive do I need, or what kind of legacy Mac would help?
Many USB floppy drives will not read double-density (720 / 800 KB) disks at all. Most that can will only read 720 KB disks formatted for IBM PCs.
I am not aware of any standard USB 3.5" drives that can read the 800 KB double-density Mac format. They are unlikely to exist as 800 KB disks as used on the classic Macs use a proprietary on-disk format requiring variable speed drives.
If that is the kind of disk you want to read, any pre-iMac Mac with a standard 800 KB or 1.44 MB Macintosh drive should be able to read them. But classic Macs are about the only system that can easily read them.
With machine like your Classic II which has a 1.4 MB Mac drive, if you can get it running, you should be able to image the 800K disks using a utility like Disk Copy, then copy those image files to a 1.4 MB floppy. The USB drive should be able to read a 1.4 MB HD floppy. That will give you an exact copy of your disks, hopefully.
Another option would be to consider something like (with no particular endorsement) the Kryoflux, FluxEngine, Applesauce or similar. They do very low-level magnetic imaging of the disk's surface, and then use hefty modern software to reconstruct the bytes. But these are mostly hobbyist-level projects and would be somewhat involved, I think.