You should be able to live full time in a RV for about the same cost as the same lifestyle in a small town or suburb in most areas of the country. The cost should be much less than that lifestyle in places like Manhattan, Chicago North Side, or Los Angeles valley where astronomical real estate values make for really high rents.
For people I know RVing, people I've met in my travels, RVing works for lifestyles as low as $2000 a month, choosing some of the least expensive places to stay. I've met working families living full-time in $6 a day municipal RV parks; it is unlikely they would be able to find a house to rent anywhere near that cost. But at the other end, there are really prime RV resorts that might cost more than your whole budget, just for site rental.
I know I could live FT in my RV for less than the $4000 a month I live on now, with no adjustments in lifestyle. Somewhat less, if I wasn't paying a couple of mortgages, and for TV and landline services I would not use RVing.
Travel from place to place, however, is an extra cost. If you are going to keep moving as far as possible every day, it can be easy to spend $300-400 per day on just fuel, not considering tire wear and other mileage based vehicle costs.
If you are going to make full-timing a continuous road trip, moving 300-600 miles every day, eating out for most meals, paying fees to visit places, your costs could be much higher than your budget, and you could be pushing at the threshold at which the excess fuel costs make RV living more expensive than just traveling by car (or bus, train, air) and staying in hotels. I travel a lot, budget $10-$20K a year for it, and each trip I have to figure out "Is this going to work better in the car, flying, or RVing."