Looking to Plus Up an Old Tenor Sax

I’ve got a, possibly 70’s-era, Bundy by Selmer. It’s been great to start learning on again, but I’d like to jazz it up a bit. I had it set up 10 or 11 months ago with all new pads and corks and etc. It still is nearly a half-step off on many notes, so I’m wondering if there’s a different neck and/or mouthpiece or reeds I could buy (maybe not $400 for a mouthpiece, but something) to help with the tones being off. Or, iffn anyone knows if good exercises to correct the bad tones. (I’m 100% willing to accept user error on the bad tonality.)

Thanks all. I appreciate you.