There never was a large population of French settlers in Arkansas. The only real settlement was Arkansas Post which was founded in 1686. The Post was abandoned several times though, with the population moving to and from Escores Rouges as necessary in response Chickasaw raids and flooding. The 1791 census of Arkansas Post showed 151 people in 27 households: 21 French, 5 German, and 1 Spanish. And that was the majority of European population in the area. It wasn't considered good farming, virtually everyone was either a trapper, trader, or soldier working at the Post. So there was a small impermanent population of French in Arkansas during the French period. It really wasn't the kind of place that was likely to develop a unique dialect or culture.

By the time of the purchase in 1803, European settlement in the future state had swelled to roughly 500. And most of these new settlers were coming from the new United States. So already by the time of the purchase, French speakers would have formed a minority in the area. And the next few censuses drive the point home. By 1810 the population had grown to 1,062; by 1820 it was 14,273. Within 17 years of the purchase the French speakers would have comprised around 1% of the population. It's likely that whatever unique dialect and/or culture had developed was gone by then.

Kansas didn't even have that much French settlement. There was a "trading post" at or near Kaw Point at the mouth of the Kansas River. But that was more or less an agreement that the French could come into the Kansa village and trade there. It wasn't an actual French settlement, and there was likely rarely any French actually there. When Louis and Clark came through in 1804 they met some French trappers in the area and saw the remains of a "Kanzas" village. But the area didn't even have any permanent population anymore, let alone a French one.