A vehicle that can be towed 4 down is a big advantage, hitching and unhitching is easier than hooking up a tow dolly or trailer and strapping the car to it. Plus you don't have to worry about finding room to store the trailer or dolly at a campsite.
4 wheel drive is a plus to explore the back country roads and trails. It doesnt have to be extreme, something like a small SUV with a transfer case that can be put into neutral or a manual transmission can be towed 4 down in most cases.
Fulltiming can be as frugal or expensive as you make it. You may not be able to afford $50 a night luxury RV sites every night or travel several hundred miles every day, but that's being in vacation mode, not fulltiming.
There are many ways to stretch the budget without having to camp host or workcamp, unless that's something you'd like to try. When you get tired of the East Coast, the desert southwest offers (generally) warm temperatures and low cost or free camping during the winter months while the rest of the country is snowed in. Then follow the weather the rest of the year. Like you said, stay for a week (or a month) in one spot and get reduced camping rates while you explore the local area. Don't forget you won't be sightseeing every day, you'll still have to do chores like housekeeping, food shopping and laundry, as well as leave time for other pursuits and hobbies.
Check out cheaprvliving.com and their YouTube videos. While that style of fulltiming isn't everyone's cup of tea, they show how people are successfully fulltiming on extremely small budgets, some as low as $500 a month. And no, it's not stealth camping in a city and hoping you won't be rousted in the middle of the night.
Another example are the WINs. They're a group of singles that are very resourceful at low cost touring. They find free or low cost places to stay for 4-7 days at a time and explore the local sites before moving on. More information including details about their current and upcoming circuits are at rvsingles.org