I had a similar problem. We bought a Jay-co 23F trailer from a dealer and they looked at our tow vehicle and said no problem. We have a 2001 GMC Yukon with the 5.3l in it.
We hooked up and left for our first trip and found that the truck was working very hard to pull itself down the road. Now in my area we have a ton of steep hills to climb, the Yukon would climb the hills with the trailer but it took everything. After replacing three fuel pumps and two ac condensers we started looking for a tow vehicle. We found a young kid that needed to off load a 03 ford 6.0.
This was the best thing we could have done. The Ford will pull the trailer fully loaded up anything hill, in fact I now pass other towing up hill.
The lesson I learned was the Toyota Tundra may have pulled the Shuttle but it was on flat ground with no wind.