Update! I feel a bit dumb now....
Thanks to the manuals, some photos and a helpful video on another forum, I discovered that the previous owner probably had the incorrect needle size. I lowered the needle by 5mm or so and it seems to work!! Now I feel a bit dumb...
As I understand, the machine should take System 88 needles, so I have ordered some to try. And the shuttle is still quite worn, so I will see if the Singer 29k ones fit, as a few people reported having success.
But otherwise this should work to start out with!