I've never been there, but I would consider the Green in Utah and the San Juan in New Mexico. Just depends on the route you want to take and how far out of the way you want to travel...
What part of Texas are you moving too? The gulf area has some great speckled 'trout' and red fish that can be caught on a fly. You can often sight fish for them.
Most rivers in texas look like a mud flow but there are few that hold trout and are quite pretty. The guadalupe is one, I forget the others.
Of course you'll have to try your luck at bass fishing while your there. And if you've never seen one, you really should try to catch an alligator gar. I've seen these that are over 12 ft long (think fresh water baracuda - with a nearly bullet proof hide). There is also some big crappies down there. And of course, there are monster catfish - but good luck with the fly rod...
Oh, and keep an eye out for the fire ants, rattlers, cotton mouths, copper heads, scorpians, and other nasties that like to hang out in the good fishing spots... Believe it or not, when bait fishing for catfish, i've hooked tons of turtles (watch out for the snappers) and even a gator or 2. Gee, now I know why I like WA so much
Have a good trip!!