I second (third?) the choice of a bit with a Billy Allen or Myler type mouth. Even one with a solid ported mouth would be better than the TT you're using now.
Here are a few options that are nice and mild. I've used either these exact bits or ones that are very similar with excellent results.
My mustang, Dobe, really likes this bit
This one is more like a mullen mouth
And this one, the picture and the price make you automatically think "Oh, it's a piece of crap", but it's actually one of my favorite bits.