If you're going to compete, get a belt and learn how to use it. IMO it doesn't really help you learn how to brace better - it just increases the intra-abdominal pressure which does help.
It sounds like you might not be breathing with your diaphragm too well, because if a weight on your shoulders stops you getting big air I suspect you breathe raising your shoulders. Try placing your hand flat on your stomach below your sternum. As you inhale, push your hand out. Your shoulders should not move at all.
As far as I'm aware in powerlifting and among strong people there is no stigma about belt use. I started out only using a belt above 85% of my max, which at the time was around 300 lbs. I now put on a belt when I decide I should, which could be on my last warm-up, or my first work set, or my last work set. It all depends. On average, I probably still stick around the 85% rule because that's what works for me. You may find differently, but I'd suggest starting with something like final warm up set, belt goes on, if you're using over 85% of your max for work sets.
There's nothing to feel silly about. A belt is a tool, arguably one of the most important and useful tools in powerlifting, and to learn to use it properly you need to use it.
I'd also highly recommend either a lever or single prong. Inzer make nice ones, as to Titan and EliteFTS as far as I'm aware. I recently bought a Harris Stability Systems belt which I'm very pleased with, but that's an Ausse company so cheap for me but if you're Stateside maybe not so much. I'd certainly recommend getting a good one, and probably a 13 mm one since you'll want one eventually.
Also consider knee wraps if the federation you'll compete in allows them (they take a while to learn too), and definitely invest in knee sleeves if only for comfort.