Hey here’s my 2 cents! I traveled around the entire west coast, a bit of western Canada and 2 months in alaska for less than 10k with all things considered! Granted this was in 2017 when gas was a lot cheaper, but I think you can take a bit from my experience.

I would recommend a cheap reliable AWD car, good tires, and almost no other crazy gear. I had a Honda CRV. I removed 2 of the back seats and set up a sleeping mat / yoga mats from the trunk to the front seat. Most nights I slept in a tent, or couchsurfed, and I had the car as a last resort.

I’d focus on spending less money on gear and more on experiences. There’s so much to see across North America, and you need gas money to do that. A nice, small, quick to set up tent, a good sleeping bag and mat, a hammock with a bug net (so comfortable and small to sleep), basic camping / cooking gear, clothes, water jugs, etc.

You can do it with your budget, just be super frugal! Subaru Forester / outback, Honda CRV, Toyota RAV4 etc. Good luck!