Sleeping in your car in either Detroit or Chicago, I would have to agree, is not a very smart thing to do. However, there is no need to go for a hotel room, where as a single, you are essentially paying for two.
I suggest you check out hostels in both cities. There are a couple of nice ones in Chicago - I loved the HI. The other at which I have stayed was nowhere near as nice, but still safe and quite a bit cheaper with a full free breakfast in the hotel dining room (If you're smart you can even make a sandwich or two for lunch.) Even though there is only one hostel in Detroit, it has some good reviews. You will be a lot safer in a hostel, even in a dorm room. Carry a medium sized padlock for your locker.
There are many other great hostels throughout the US. Most have a breakfast, all have visitor information, many run tours to local attractions and lots have discount tickets available...... sometimes even free.
In the Moab area I would highly recommend the Lazy Lizard - a great hostel.
There are usually cupboards of free non perishable food available, and usually there is tea and coffee 24 hours. Besides that you will meet lots of other travellers and be able to share stories, plans and ideas. Hostelling is a fun and affordable way to travel long term.