I made a transition on alto from a Selmer Soloist D, using Rico Jazz Select 3Soft reeds up a couple of sizes to a F opening not too long ago, and had to make the trade-off of going to a softer reed, RJS 2Medium or similar. It didn't take long at all to get used to the wider tip opening, the plus side being more volume and better pitch bending. I'd had a wider-opening Beechler Diamond Inlay (it has 8 embossed on it) which I'd always found a very hard blow until going to the Selmer F, but now I've married up the Beechler to the Dolnet alto it's sounding and playing fantastic.
Similarly, I've gone from a Selmer C* to a Jumbo Java T75 on tenor (I think this is the widest of the Jumbo Javas) and again, once I got used to it in a couple of days the tone and volume are great on my old YTS-23.
What can be hard, after moving up to a wider opening, is going back again - not too much adjustment back to a Meyer 5 or Link 5 on the alto (although noticeably quieter), but going back to the Selmer C* on tenor was horrible. Much quieter, felt very resistant, loads of bum notes and squeaks, and the tone was kind of strangulated - so awful, in fact, that I thought something must be wrong with the sax rather than me.
Going back to the point about different pieces for different styles, I suppose it's true to an extent, although what shapes the sound most is your own playing. For that reason, although a mouthpiece may have 'jazz' as part of its moniker it doesn't necessarily mean it'll only be suitable for jazz. I'd recommend trying out a few side-by-side, and you should find something that you feel comfortable playing and that gives you a sound you like. I've found in the past that I know almost immediately if a mouthpiece feels right for me, and playing it for only a few minutes tells me whether I like the basic. After that, it's down to seeing what I can get it to do.
Cheers, Bob