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In the Southwest there is not shortage of scenic drives. That loop seems approachable in one month's time, with the noted stops.

As for camping for free, something that comes up here quite often is sleeping at truck stops. But I wouldn't bank on being able to find the typical camping site for free. Which brings up a question - do you think that your budget will cover all of your expenses? If so, great - but if not, then I would question the feasibility of completing this trip.