Simon it's a fundamental principle of saxophone construction that all components do a job
If you follow this principle when a piece of cork is missing you can expect consequences
It's quite difficult for us to advise you on the mechanics of your sax without pictures. If you leaf through the website you'll see most technical questions are accompanied by visual clues. I don't know of a blind saxophone technician.
Can I second the suggestions in other posts that you find a friendly tech to take a look at your horn and tell you what needs done? In another thread I recommended sax technique literature. In this thread I can recommend Steve Howards most excellent instruction manual and his entire website for good measure?
Good luck in getting your soprano set-up just right