Soprano Sax G-C tuning issues.

Hello everyone, I've been playing soprano and alto saxophone for roughly 12 years now, and I have recently noticed that on my new soprano, my G,A,B and C are extremely flat. It is too the point where they are only in tune when the mouthpiece is completely covering the cork. And by that point every other note is super sharp. I'm specifically talking about C5 and down. Around F4 it seems to be in tune, but everything between those two is sloppy and super flat. Any suggestions? I ran a light through it and my only leak is a super tiny super low Bb.