Yes, big league pitchers use them. I’m not aware of any MLB organization that has an organization-wide weighted ball program, but individual pitchers use them throughout the bigs.
For me personally, weighted baseballs (and swimming) is strictly an off-season activity and I’d limit the overload/underload to 6 oz and 4 oz. That’s what made me successful. But others may have something else that works for them – I’ve seen pitchers use much heavier weighted balls.
Weighted balls aren’t just limited to big league pitchers. I recently spoke on the phone with Alan Dunn, the pitching coach for LSU and a former Chicago Cubs coach of mine, and he said one of his pitchers is using the WB program I have in the Articles section on my site. Dunn said the pitcher is a fringe guy who needed to try something different to break through a velocity plateau.