As Michael points out, your budget is not that bad. However, there are many ways to save money on a road trip, and sleeping in your car is not the best of them. It may pay you to spend some time reading through this forum.

Along the way you will come across rest areas and truck stops where the hotel/motel discount coupon booklets are available. Check these out, there are some good deals to be had. Mostly they are for walk-ins only.

Whatever you do, do not purchase your food and drinks in convenience stores or at fuel stops. Shop in regular supermarkets. Many of these have excellent variety of salads and/or meats available for a good meal.

To safe even more though, I would second the suggestion to get yourselves a tent, sleeping bag and mats as well as a cooler to keep drinks cold and sandwiches for lunch. Even if you camp every second night, the money saved should be more than the cost of your equipment. State Parks and Forests as well as BLM lands have campgrounds which are generally around $20 - sometimes cheaper (even free). These are marked on good maps as little triangles or tents.

One worthwhile expense is to make sure you have a roadside assistance package, such as provided by AAA (even the best maintained vehicles can break down) where you can also get good maps to help you in planning. Be sure to carry maps when on the road. An alternative is a good road atlas. Don't be tempted to rely solely on your electronics.

The link above has many more suggestions. Enjoy your trip.

Lifey