Not at all. Very legitimate issue.
I'm new to diving; I've done one PADI resort course, so 2 hours in the pool and one OW dive of sorts, to 11.8m. I noticed that I had to pay a fair bit of attention to keeping the reg in my mouth, and I wondered if it's possible to buy a mouthpiece that makes that easier. I have a pretty decent overbite and a small jaw overall, and felt like I had to keep adjusting the mouthpiece as it tried to float away up to the right. It was do-able but distracting.
While 'armchair diagnosis' from afar is always dangerous, your post suggests (at least) three things might be at play, and I offer these in no particular order of likelihood or importance:
1) New diver syndrome. Many new divers are particularly anxious about making certain that their mouthpiece stays in their mouth, they tend to bite down a bit harder on the mouthpiece, and are more sensitive to any small perturbation of the position of the mouthpiece. So, the second stage wanting to float up to the right may have been both somewhat real, and somewhat perceived, enhanced by concern about losing the mouthpiece. That is by no means a criticism of you or any newer diver, just my observation of students / newer divers, as well as a fairly vivid memory of my own early diving experience. If that was part of the issue, the good news is that it tends to improve with time and dive experience.
2) the described sensation of the mouthpiece wanting to move to the right could reflect a second stage hose that was just a bit short for you / your rig, or it could reflect a second stage that was a bit bulky / heavy. Or, conceivably it could even reflect a hose that was VERY long and was producing some drag as you moved through the water (the comment about the second stage wanting to move 'up' brings this to mind - excessively long LP second stage hoses can be some 'floaty');
3) a lot of resort regs have very cheap / flimsy / excessively flexible mouthpieces on them. And, they don't accomodate an overbite very well. I agree with the several comments suggesting you try a Seacure moldable mouthpiece. It can be trimmed to fit your jaw depth, and molded to fit your overbite. I have a slight overbite as well and noticed when I was first certified that I had to adjust my jaw position to get a good grip on mouthpieces. My first reg was a Zeagle, and I did not care for the mouthpiece that came with it and changed to a Seacure early on, which I really liked. An alternative mouthpiece to at least try is an Aqualung / Apeks Comfo-Bite. After a number of years of diving, I happened to start using one of these when I bought an Apeks reg, and I LOVE IT. It is modest in size, but fits my mouth very well. After using it for a while, I went back to my Seacure and found I actually did not care for how bulky it felt. I subsequently changed all my regs to Comfo-Bite mouthpieces, including the Zeagle with the Seacure (which now sits unused in my gear box, as a spare). Personally, the Comfo-Bite is the most comfortable, best-fitting mouthpiece I have ever used. It may not be best for you, but is worth trying.
I will likely be renting gear for some time, although if buying a reg would make everything more comfortable I might be able to swing it at Christmastime.
I suggest you consider it, if you feel that diving is enjoyable enough for you to anticipate diving multiple times a year. If you see yourself as a '1-2 dive trips/year at most' person, the expense may not be justifiable. I think a reg is a good early purchase, as it is a 'personal' item in the sense that it gives you a degree of comfort and confidence in your air supply, and you become familiar with how it breaths, for YOU. The recommendation for an Aqualung Micron is also a good one. It might be that the best solution for you is a custom mouthpiece, a second stage hose that is the optimal length for you, AND a second stage that is light in weight. While I personally find omniswivels very useful for certain situations (my sidemount regs, for example, both have them), they are a bit lower on the list of things to try in my opinion.
Is it possible to just buy a different mouthpiece to use in rental regs?
Yes. The comment about the Seacure is important - one size does NOT fit all. But, you could conceivably buy 2 Seacures (I and II; I wouldn't bother with III), fit them each to your mouth, and take them along. That would be cheaper than a reg purchase.
