I'd stick with one character per run unless you're bursting with confidence and prepared to be all over the place.

We have one guy who comes and goes with a new (or returning) character every 2 or 3 months and he's fuckin' amazing every time. And of course, several guys who create multiple characters.

I think you're doing good here though. You're someone who the GMs and your partners/opponents can depend on to give a show your best effort. That goes a long way, and that might be why you're having other handlers interacting or "attacking" you from so many directions. Take it as a compliment, but yes, don't hesitate to let them know that you're tied up with other commitments.

That pretty much goes for anyone. If you're getting attacked or called out on the shows, it means that the person coming after you knows that you give a shit, and they are interested in going back and forth with your character. After all, we all hate RPing against someone who we think isn't going to show.

If you don't like how your character is being booked, let the GMs know so they can adjust where they place your character.