Interesting Question. I must confess, I'm completely unfamiliar with Outlander, however, I will do my best to tackle this question from a historical perspective. Also, I am not qualified to discuss slavery in the 19th century, so my answer will focus exclusively on 18th century slavery. I hope someone with background in 19th century slavery in America can shed some light on this for u as well.

First, the questioning of what is 'possible.' This is drastically decided by time and location. Generally speaking, colonies and states took steps to make freeing slaves more complicated as time passed through the 18th and 19th centuries.

Still, many southern states viewed freed slaves an an "inconvenience" and took steps to prevent or hinder the ability for someone to easily free their slaves. For instance, in Virginia, the local government took steps to prevent colonists from the "ill directed generosity" of freeing their slaves. In 1691, the Virginia assembly enacted a law that allowed people to free their slaves, but mandated that all free slaves leave the colony within 6 months and the financing of that trip had to be paid for by the owner -- making it a financial burden for most slave owners at the time.

Virginia revised that law in 1723, when they passed a new law that made it much harder for slave owners to practice Manumission, the legal process of freeing one's own slaves. The law itself said:

No negro, mullatto, or Indian slaves, shall be set free, upon any pretence whatsoever, except for some meritorious services, to be adjudged and allowed by the governor and council, for the time being, and a licence thereupon first had and obtained.

Essentially, the Virginia government made it so that no one could manumit their slaves without the government's approval. Worse, was that the 'meritorious services' stipulated in the above law did not apply to situations like in Outlander where someone was saved by their slave. Instead, it was more often than not, when slaves informed their masters or the government of fugitive slave locations or discussions of slave rebellions. By some accounts, fewer than 25 slaves in Virginia were able to earn their freedom between 1723 and the American Revolution. This was reversed in 1782, but it did complicate things further because, "The law also mandates that anyone manumitting their slaves shall provide support for those over or under a certain age and that slaves pay the taxes and levies required by the state." This meant that slaves who were still deemed not be adults, generally 21 or passed retirement age, were required to be paid for by the owner, via a stipend. This made owners less likely to manumit them since this now could be creating decades of finances to pay their former slaves, when much of Virginia's population was experiencing an economic decline.

Now, while I stated all this, it is worth noting that freeing slaves was much easier in states north of Virginia than those below it. Thee types of manumission prevention laws were not nearly as common or complex in other states. For instance, in Maryland, Quakers began talking about manumitting their slaves on moral grounds as early as 1763, with many communities completely abolishing slavery in the state by 1778.[1] Such actions would have been nearly impossible if they were proposed in Virginia.

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So to sum up, it would have been extremely hard for people to manumit their slaves during the colonial period in the South, while it would have been more possible in the North. Things also changed in states like Virginia where prohibitions against manumitting slaves were reversed during the American Revolution.

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1] Aglietti, Jason B. The Friends They Loathed: The Persecution of Maryland Quakers During the Revolutionary War. Proquest LLC., University of Maryland Baltimore County. 2018. pp 15