Sharon, your Suburban should pull the 31 footer just fine. If you got your wiring and brake controller thing figured out, I guess you have a weight distributing hitch too. Hopefully you have access to somebody that knows about these hitches to help you properly set it up. With a trailer that long, the air off passing trucks and buses may cause the trailer to sway. If that should be the case, you will want to look into some kind of sway control device before you start your tour around North America.
We towed a '98 31 footer with a 99 Tahoe for a couple of years. That trailer weighed 7,000 empty with an 8,300 max. I don't know if your trailer weighs the same or not.
Good luck, be careful, and have fun!