When we were "shopping" for a RV, I used Google a lot and found a large number of places that offer "used' RVs. I would suggest that you invest time in shopping, then invest even more and when you find the floorplan and equipment that you need, structure your trip around the location of the RV. Also, consider the task of selling the RV at the end. The seller might be willing to re-purchase your RV when you are ready to go home, especially if it is a large RV operation, i.e., high volume. There's probably one of these in LA but there are many others, Phoenix, Az; Houston, Tx; Florida; etc.
Check out the tax implications of the purchase location/delivery. Some states have very high sales taxes and others do not. Some states have high registration/licensing fees, some don't. For three months, renting is a true option; for six months or longer the scale tilts toward buying, I would guess. Especially right now, since the costs have dropped so much compared to a year or two ago.
Your comment about studying maps is also excellent. Maybe you have the "endurance" to drive 600 miles per day but I think that would be a real burden and get old really quick. If you plan on a few long days, fine but I believe you'll enjoy the trip far more if you pick some places you want to see, go to these places and plan to stay there for several days, then go to the next place. It might take several days to get from one destination to the next but 300-400 miles/day is a lot, IMHO. Also when you are getting 10 or less MPG, fuel costs are a real factor.
There are RV "tour companies" that help with all this planning and logistics for a fee of course but it does give you a guide, especially if this is your first visit.
As others have mentioned, I hope you come and have a great vacation. There is so much to this country. Stay in touch via the forum, and if you suddenly need help, ask. Good Luck.