Saxophone repairman Matt Stohrer of
https://www.StohrerMusic.com talks at length (again) about a handy tip for soaking rusty/grimy pivot screws in Zep Citrus Degreaser to make the job of cleaning them a lot easier.
For modern horns or clean vintage horns, this will likely be all you need. I have heard of people doing basically the same thing with other solvents, but this is the one I use so I figured I'd share.
If this treatment doesn't do the job, you may find this video helpful:
Saxophone Repair Topic: Cleaning Rusty Pivot Screws
https://www.youtube.com/watch?v=p7lCe_WQZQI
If you like my content and would like to see more, you can support me on Patreon or by "buying me a coffee"!
Patreon:
https://www.patreon.com/mattstohrer
Buy Me a Coffee:
https://www.buymeacoffee.com/mattstohrer