I just switched to New Teams a few days ago. And I just had my first meeting where I tried to share my screen. It was a Visual Studio window. It almost immediately stopped updating for other attendees. I tried to stop sharing, but only the share dialog showed up when I pressed that button, as if I wasn't already sharing (even though it was apparently still sharing.) I tried leaving the meeting, but for 10 minutes now, the meeting Window has been stuck open, saying "Leaving..." I can close that Window, but when I try to Join again, I get the same "Leaving..." window. Seems I'll need to restart Teams (and then switch back to Classic).
A pop-up appeared at some point, "We've detected a problem. Try again in a few minutes. " -- or similar.
In short: My first experience with screen sharing in New Teams was a disaster.
My observations: New Teams is a lot faster and uses fewer resources. But that's apparently because it doesn't work!
I won't be included to try New Teams again anytime soon.