Do US universities validate your degree before making an offer?
Some universities do, some don't. For those that do, it is not because of suspicion on the part of the search committee that you don't really hold a degree. The academic world is small enough that we rely on mutual trust for many things: presumably the committee members know some of the people who wrote your letters of recommendation. However, especially for public institutions, there are often various rules and regulations that must be followed, paperwork to be filled, and other details to process. The fact that you have a degree from a foreign university is probably slowing things down, as the admin staff may not be used to dealing with it. On se débrouille.
If you really like, you can email the search chair, asking if there have been any updates, as you are anxious to get a firm offer in hand. Also, remember, a verbal offer is worth the paper it is written on. Until you have signed a contract, you are still on the job market and you should be acting as such (e.g. applying for jobs, attending interviews, etc). However, as you've said, there's a lot of uncertainty. So... Relax. Take a walk. Write that paper you've been procrastinating on. Enjoy the long French evenings.