Hello and welcome.  Years ago I was in a similar position to yours, with a couple of school years worth of clarinet lessons more geared toward raising band fodder than developing good technique.  During the third year I deided I'd had enough, then a decade later I decided to go for my long-term dream of being a saxophone player.  My budget and schedule didn't accommodate lessons, so I picked up what I could by hook or by crook.  You are fortunate to be embarking on your venture in the age of online lessons.  I'm sure I could have developed better habits earlier on had I started out with some lessons emphasizing proper embouchure, tone building, breath control, and voicing.  I don't know what instruction you had in those areas through school lessons, but from what I've seen and heard it might be totally inadequate.  For example, some instructors teach the rolled-in lower lip embouchure for saxophone, which gives a quick sort-of-ok result but leads to big problems down the road.  You might be better off considering yourself as starting from scratch as far as technique is concerned.  At least a few one-on-one lessons, either live or through skype, might help you a lot.  Google "online saxophone lessons" and you'll find something for every occasion.  

Here's a site that I've found full of useful tips and the "bulletproof saxophone playing" manual sold on that site is also good.  

http://www.bestsaxophonewebsiteever.com/category/best-saxophone-tips-and-techniques/

 If you feel a need to replace the mouthpiece, that's something best discussed with a teacher who has heard and watched you play.  If your current mouthpiece doesn't work out, the Yamaha student mouthpieces are good.