Here are some trustworthy official sources (in French however):
Briefly, you need to show form Cerfa n°10070*02 and two copies of the full list of all effects you want to import to the customs with their estimated price (being able to present invoices can be useful for that). You will keep a copy of your inventory with a stamp or something from the customs administration (“exemplaire visé de l'inventaire”).
You also need to establish you have been living abroad for the last 12 months and that France will become your main residence from now on but there is no set list of documents you need to present for that. Things like utility bills (for the US), a job contract or a long-stay visa (for France) should be good but the official sources are vague about that, presumably to leave some discretion to the customs administration. What you read on the web is probably what worked for other people before.
You can import your things in several times over the course of 12 months after moving to France but you need to present the full list the first time you show up with goods that you want to import under this rule. Presumably, this means you could have the rest sent over by post, but only after contacting the customs with the form and the list. Enclosing a copy of the certified inventory would seem like a good idea in case the customs take interest in your package.
Note that alcohol, tobacco and goods of “a professional nature” are excluded from this exemption. There might also be other rules and restrictions (unrelated to duties) for things like plants, pets, food, weapons, gold, or cultural goods either in France or in the country you are leaving. Finally, there are special rules for cars, boats, etc. but your question suggests you are not planning to import one.