Auto Europe will help you find your options and shop for whatever you need. They're a US-based brokerage with a good reputation. You can use the website to search and then see if you can get a better rate from the company you choose, but I've never seen that happen. Booking through Auto Europe requires prepayment but it's easy to cancel or change your plan. I booked a car with them months ago, then revisited the site to find that the price had dropped considerably. I called them and got the lower price, also upgraded the car, and got a refund on my credit card for the difference.
Your plan is certainly reasonable, though you'll probably incur a dropoff charge for returning the car to a different location. That may be worth it, rather than driving back to Nice and then taking a train or bus back to Avignon or Aix, costing time and money. Many train stations have car rental locations, as do airports. You should know that French law requires you to carry a translation of your state-issued driver's license -- the simplest way to provide this is with an International Driver's Permit, available at low cost from AAA offices throughout the US. You'll need a photo, or they'll take one for you.
De rien.
Edit -- Note the opening hours of rental locations when you make you plans. Many are closed on Saturday afternoons and Sundays, some are closed weekdays for a lunch break. Auto Europe has this information and hopefully won't let you book for timeframes that won't work. Double check with the rental company's website if in doubt.