The straightening isn't really a problem. When you start, you SHOULD be straight up. You're gonna get hurt if you try to start leaned over. 5 ft later, with both feet on the pegs, you should be able to turn. Even on a right turn, you still have at least a half a lane, and you're only going 3-5 mph.

I think that's a drill I'd do if I were you. In a parking lot, start straight, get feet up, then do a turn. If you're going less than 5mph, you may be able to turn into it, and give a little throttle to keep it from falling inwards. Above, it's going to involve countersteer.

See, just when it seems to make sense, it turns out the handlebars cause 2 different actions depending on speed.