It looks like your injury is fairly recent and you're still learning the ropes. I'm T5/6, so I don't know if my own experiences will line up with yours. To that end, if anything I say isn't helpful just ignore it. If it is helpful, that's even better.
I'll repeat what everyone has said. Get yourself a proper wheelchair fitting! If you can't find a PT that has the experience, then scour the internet for information. I imagine that if you take some pictures of what you're using now along with pictures of you sitting in it, some folk could give you feedback.
When I was first injured I had exactly the same problem. I had problems with falling forward and I also had problems with side to side stability. I was a human weeble wobble! Anyways, my original chair was fitted with swing away lateral supports and I wore a chest strap as well. But, eventually, I found they just got in the way. I think it's a pretty common experience that everyone figures out how to compensate with the muscles they have left. With enough work, I have no doubt that you can do the same.
Here's an exercise you might want to try. Sit at the edge of your bed or a PT table without any back support and play catch with someone. I promise you that, at first, it will cause you to lose balance and fall backwards (you might want to put a pile of pillows behind you just to be safe). Start with a balloon and have your partner throw it to you at about chest height. Once you can handle a balloon switch up to something a little heavier (e.g., a beach ball). Once you can catch the beach ball switch up to something heavier after that. As you get better, your partner can even start throwing the ball to you at different angles which can force you to reach.
Finally, one thing that might be getting to you is fear. When you lose your core muscles what used to feel like a little lean feels like you're taking an enormous dive. I remember when I was learning how to tie my shoes, I was convinced I was going to fall out of my chair. As with all things, you adjust.
Graet advice, stephend I remember my rehab PT setting up the balloon throwing exercise some twenty years ago, and me thinking how stupid it was going to be. How difficult could it be to swat a balloon, after all? Was I surprised!