Factual error: After deciding not to cross the Red River, they go west. Shae warns of two things: lack of water and having to cross the Brazos River. But to the west, there would be plenty of tributaries of the Red River to supply water. And since they would be traveling west between the Brazos and the Red River, there would be no need to cross the Brazos. Further, if they did cross the Brazos, they would find themselves south of the Brazos, where they would be forced to turn around to cross again.
The Crossing - S1-E4
Factual error: The Dutton camp as well as Shea Brennon and Thomas are using modern thin Teflon/non-stick coated coffee mugs/pot.
Factual error: Horses tied behind Elsa seem to be wearing nylon flat halters that weren't invented until Feb 1894.
Factual error: The bird Elsa is talking to when she died is a Eurasian (ring neck) dove, not in North America till the 1980s.