Okay I have gone through quite a few hoops, but am currently trying to make a “backup file” using ddrescue. A few read errors popup, so there is at least something faulty on the sd card. In fact, I ended up trying ddrescue because on win32diskimager on my windows pc wouldn’t do it. And trying to make a backup image using dd on a orange pi zero I had lying around (I’m pushing that tiny thing to its limits like this I suspect) also ran into an error around the same size as win32diskimager. The biggest problem right now is trying to copy the partition of the 64GB sd card to a 64GB usb stick, since I had nothing bigger lying around. And the writeable part of the usb stick is just ever so slightly smaller 
The command I ended up running is ddrescue -f -s52GiB /dev/sdb2 /dev/sda1 mapfile, with the -s flag being there to limit the maximum output size (I believe). There’s only 32GB of actual data on there, so hopefully not too much gets lost/corrupted, although I should have a new card arriving tomorrow.
Posting this mainly in case someone is running into similar problems. The backup is just in case the sd card does fully corrupt before tomorrow. For the new SD card I’ll be going with a clean install of raspbian, since mounting the partition of the current one on a (different) linux machine also gives access to the base linux file system.