I have a 2003 Silverado ext. cab. 5.3l and 3.73 rear end with auto trans.
Within the past 4 or 5 months the rpm has increased to aprox. 2200 while
driving highway speeds 70 - 75. Before this it ran around 1600 -1700. It
feels like the truck is working against itself and it wants to shift to another gear. Just had the computer checked and there are no codes. Any ideas ?

When was the last time you had the trans serviced? Perhaps the fluid isn't as viscous as it use to be? I'm shooting in the dark here....

The truck has 38k on it.

Past due for a tranny flush, IIRC

Past due for a tranny flush, IIRC

I'm pretty sure my manual does not have a scheduled tranny flush for quite a while past 38,000 miles unless it's used as a taxi, super humid area, or some other extreme condition. Reading mine, if you just do hwy driving with light loads and no or very little towing like me, you never have to change it. Of course, I think that's crazy, but it's what my book says. Doubt it's the fluid.

One obvious thing to check is the tow/haul settings. Maybe it's not shifting into high gear at all. That's the RPM range mine stays in at hwy speeds when tow/haul mode is turned on.