Claire and Jamie are married and hand-fasted in an early season but when Roger and Brianna hand-fast, Claire acts like she knows nothing about hand-fasting.
In season 6, which takes place primarily in 1773, the discussion over the coming conflict with Britain is mainly framed as being about independence from the king. However, in the early 1770s most Americans did not see the issue as being one of independence but rather securing the rights of Englishmen. Furthermore, they persistently portrayed their conflict as being with Parliament rather than the king. The problem as American saw it was parliamentary overreach, which threatened their liberties. Americans only really fully turned against King George once hostilities broke out and it became clear that the monarch supported the use of violence to maintain order in the colonies.
The Jacobite cause, at least in Scotland, is persistently portrayed as a nationalist one with nationalist conflict between native Scots and an alien English military being a primary driver of conflict in early seasons. In reality, the Jacobite cause was mainly a dynastic conflict about the proper role of monarchy in government, as well as a conflict over parochial rights and causes. Nationalism didn't factor in, and while many Scots, especially in the Highlands, were sympathetic to the Jacobites, many others were loyal to the Hanoverian monarchy of King George.
In the seasons set in America, Claire and Brianna are frequently shown bare-headed. In late 18th century British North America, women would have been required to have their heads covered while in public. Claire's lack of a bonnet would have been regularly commented on by other characters and would have influenced their perceptions of her status and propriety.
In a few episodes, characters say"OK." An American term that didn't come into use until the 19th Century.
Nothing is said about Claire's strange accent. That which today is known as a reasonably representative British accent uses such sounds as a soft R. In the 18th century this was far from common.
Throughout the series Claire keeps mispronouncing can't.
Instead of saying it the English way, she repeatedly say's it the Irish way.
This is an actor error by Caitriona.