Though you're probably out of luck, geographically, Lee Parks Total Control classes
http://www.totalcontroltraining.net/HTML/Schedule.htmlMight resolve a lot for you.
Yes, it is possible to push too much - not too hard. Generally speaking, a harder push will change the direction of the bike more quickly. But keep in mind that of you grossly over do it you can end of causing a low side by pushing the bike too far to one side. But as said, this would be the case if you grossly over did things.
An assertive press on the outside bar as you're exiting a turn will stand the bike up more quickly than - say just going to neutral pressure on both bars and letting the bike come up on it's own.
If suggest you find an empty lot and give it a try under circumstances where there's not walls, cars, debris or other crap to avoid. Start off with a slower, larger diameter circle in one direction and then when you complete the circle give the bars an assertive press back to netural and see how it feels. The bike should right up quickly.
Lather rinse repeat a few times adding some pressure to the push progressively as you comfort with the reaction of the bike permits. You should notice that with more pressure the bike rights faster.
Work at that for a bit in both directions. Ways to make it harder will be to tighten the circle diameter - DON'T go faster. Tightening that circle just a bit will feel plenty fast.
After a bit of this, if you feel improvement, open up the circle and see if you can transition from one direction to the other (Figure
and work that.
Take it easy during these drills. It can frazzle ones nerves pretty quickly.
In addition to the bars, do not forget about your lower body. Managing your weight though the pegs will do a lot to help the bike transition from one lean to the opposed side.
And probably the biggest item, that you already made note of, is looking/vision. The bike will track where you look. So if you're in a right and know you've got a fast left coming up, you'll need to be looking left potentially before you you have fully exited from that right. If that's a problem *SLOW DOWN*