The Quiet Assassin

Remember to budget for some form of anti theft device. High tensile chain and a matching padlock as a minimum.

I am from the UK and did this a few years back.

First step is join Escapees and register in their mail program. This gets you a mailing address you use for registering your vehicle and getting insurance. 

Getting insurance will be a PITA as you are unlikely to get a US drivers licence. I found that Progressive and Foremost were the only ones who would cover me. I used  Champ Carter Insurance Agency LLC at (936) 327-3804  in Livingston Texas who are familiar with the problem of insuring overseas visitors. Read the small print carefully. You may want to go with the state minimum cover. basic insurance  Things may have changed. 

At $5000 you are going to be looking at Craigslist Rvtraderonline as most dealers will not handle stuff at that sort of price. You can definitely find something that will get you round the USA PROVIDING you can fix stuff yourself. Make sure it is built on a popular chassis. Avoid European imports.A Chevy Tioga would be a good choice. Spares available everywhere. Buy a petrolengined RV. Petrol is $2.18 in Utah diesel is $2.85 getting a diesel worked on will be difficult and expensive. Anybody can fix a Chevy 350.

Look at the tires carefully if they are older than 7 years budget for a new ones. tire age

The 1 year National Parks pass is an amazing bargain at $80 Some campsites are free with the pass.

I hope you get it together. We did it and had a great couple of years.

Remember to think about security on suitcase gennies.

3/8th high tensile chain and serious padlocks.

Ramp for the dog. We traveled with a geriatric GSD who could not manage steps but coped OK with the ramp. Get one that is as long as possible. Ours folded in half and stored across the RV.

Two good handholds, one either side for humans