Hi Mark, it's a nice sound - probably the most important thing as the Sop can sound really nasty. Wade is right, you have a good ear but, not surprisingly, you are using your tenor chops and they're a little too loose. Your musculature will improve, but you might need to consider using a little jaw help.
A decent experiment might be two-fold: put you jaw forward a little (forward, not bite). Maybe couple this with lowering the angle of the Sop too - don't let the head go down as this is the same angle otherwise.
Does this make your initial pitching above the break better?
As for keeping two or more horns going, many of the great multi-instrumentalists rotate their instruments in each practice session. And yes, I believe that any practice is beneficial no matter which horn.
You have the makings of a really Sop player so stick with it. I think that it's an easier thing to sort a bit of under-pitching than being too tight.