The Louisiana Purchase area had actually been under Spanish control for about 40 years before the Purchase. It was transferred to France, then to the Us a few months later. Having said that most of the settlers were French (mostly from France proper, not Cajuns from French Canada, but some from French Caribbean territories, and some slaves and free black/creole families as well).

When the territory was turned over to the US the population became US citizens (except slaves who were still considered chattel) and all personal property was respected, including all land titles.

Someone deep in the territory, so much so that they did not hear about the transfer until months afterwards was probably a frontiersman and didn't much care about governments as long as they were distant and not in their day to day business. Everyone else would have found out reasonably quickly and it would not have affected them much at all initially. As time went on and Americans started pouring into the territory the French people found them quite irritating and socially inept, particularly in New Orleans, but they also started making money as the economy boomed, which it did fairly quickly.

This article might be enlightening - it refers particularly to French traders on the Mississippi.

The social reality on the ground was slow to change, and French settlements continued much as they had before - tied together by a commercial system of traders and merchants travelling the waterways of North America to form what was in essence a French river world.

He goes on to say that this continued into the 1830s, buoyed by fur trade from Saint Louis, and inevitably fur trade from Michillimackinac and Canada.

Interestingly, he makes a point I wrote a paper on for a university class - the French trade networks were similar to Native trade networks, often based largely on loose kinships (godparents, remarriage, babysitting) and had a pretty significant amount of mobility for women.

That aside, the situation of the southern French should actually be contrasted with Michillimackinac, a British fort after the Treaty of Paris. The French only lightly fortified it even though it was by far the most important trading post south of Montreal, even moreso than the Sault. There was a large trade community there formed almost entirely of meti (half-Indians, children or grandchildren of French traders). By comparison, the British more than tripled the garrison, erected heavy fortifications, and eventually removed the previous population there for fear of the Indians - Indians who would actually storm the fort during Pontiac's War, having convinced the British to join them in a game of lacrosse just outside the gates.

The French really did not remain where the British were, especially with the removal of the Jesuits. Many Indians who had been Christianized continued with their uniquely hybrid faith, even after being displaced by Americans. And the meti/French trading community thrived, as did that of more southern reaches. There's a reason that Lousiana today has so many Cajuns and Creoles - because the French never left, they just became the new Indians.