That was me, 30 years ago. Taking 4-5 months a year off, driving around, mostly in North America, camping, hiking, backpacking, and sleeping in the car. Collectively, that’s a few years on the road so I have a lot of thoughts.
$250/week + fuel is easy. Half that is doable.
Buy all food at grocery stores, don’t eat out.
I buy fruits and veggies I can eat raw and canned foods (or $1 Pasta Sides of Rice Sides at Walmart) I can heat up easily.
If you want hot food, fire up your stove.
Take sponge baths in restrooms. Fill a liter bottle with warm water, go into a stall, and scrub up. If it’s a single-occupancy restroom, lock the door and go to town. Strip down completely, clean everything, wash your hair in the sink, etc. Clean up really well after yourself.
I-80 and south, McDonalds’ have multi-person restroom. North of I-80 and along I-90, McDonalds’ have single-person restrooms which allow that more thorough cleaning.
A reliable van is going to cost more than a reliable compact car. And a compact car will use 1/3 to 1/2 as much fuel as a van. I sleep well in a compact car if I bring a full-sized pillow and recline the front seat. I have a speech rehearsed, “I’m sorry officer, I got tired while driving and felt I should sleep a few hours before continuing.” No one is going to cite you for that and the only time I had to use it, he only told me to move another 5 miles down the road and use a parking lot there. Lots of folks sleep in Walmart parking lots in cars, vans and campers. Just park around the perimeter – don’t take the prime spots.
Rest areas are for just that. They say “No overnight stays” but that’s only so they can cite people who are camped out for the duration. A single night, spent in your vehicle is never a problem.
Trailheads can be a nice place to camp in the car. If someone hassles you, you say you got in late at night to start hiking the next morning. You could sleep one night in your vehicle, take a long day hike, and sleep another night in the vehicle.
Big-city universities now require ID to enter their gym, pool, etc, but in more rural areas, often anyone looking like a college student with a book bag, can wander into the gym. However, your book bag has a towel, soap, and change of clothes.
Hitchhiker’s Guide to the Galaxy was right – don’t forget your towel. Yeah, your backpacking towel, but also a nice big cotton towel for the car camping / college gym portions of the trip.
Youth Hostels aren’t as close to each other as they are throughout Europe or even in the NE USA, but they are less expensive than any hotel and a great place to connect with other adventurous travelers to share hikes, carpools and fuel expenses with.
WWOOF’ing (work on farm) and Workaway are options. A friend here in Alaska has hosted multiple groups (he’s a dog musher so there’s always poop to scoop and wood to stack). I’ve housed some of his overflow. You work a few hours in the morning in exchange for room and board and (sometimes) working interactions with locals. Look for references and reviews – a very few hosts abuse the concept, but most are great experiences.
Look into couchsurfing.org I’ve got friends that have travelled a lot of places, with no lodging costs, and met lots of interesting people. If that appeals to you, consider hosting a few people before you leave home – then you build some cred and reviews before you ask others to host you and you learn the social etiquette and norms.
If you post more specifics about where and when you’re thinking of, I could offer more detailed ideas. Also, you’re welcome to PM about air/land transport. I’ve flown over a million miles around US/Canada and driven a lot more than that and have learned a few tricks how to do it cheaply, free, and/or get paid to move around North America. Sometimes opportunities arise like someone needs a vehicle moved from here to there that could save you even having to buy a vehicle at all.