So I got the solder pump and it definitely does not remove all solder. It only removes small blobs, it doesn't remove solder that's flat.
I was able to get a cleaner area though, and I used the solder iron to make sure no solder connected between things that I think shouldn't be connected, but checking with the multimeter,
now everything touches everything.
I really don't know what to do. How does this work? Is everything connected to everything else in headphones?
Edit:
Re-cleaned solder with the iron and the shorts are fixed.
I also compared the two new elements to the one of the old pair that worked and they are the same as far as connectivity goes, so now I'll re-solder everything and... Wish me luck.