Because of the history of laws made after the Nat Turner Rebellion in 1831, my money is on Harriet being born free.
I assume that if you think there is no DNA connection to the Cohen family it is because you have scoured the distant DNA cousins in your DNA results and not found a commonality with that family, but I would suggest finding what commonality is there and dig deep into it. DNA doesn't lie. If they are linked as a cousin, they are a cousin. You are related regardless of whether you know why right now or not.
Also, if Ancestry's Thru Lines links you to any euro families, dig into that as well. An ancestor of yours may link to a Cohen neighbor or a Cohen employee or a recorded visitor.
I would also recommend looking at GEDMatch matches that are less than 7cms. This distance, looking at people born in the late 1700's/early 1800's (Harriet's ancestors would be), is right where that 7cms dips into the 3-5cm range. I know lots of people think that is an insignificant amount of DNA, but if you are researching ancestors of someone who is listed in your DNA matches, you are not looking at someone random, and that low level of matching DNA is not statistically insignificant for you. Don't dismiss it.
If you have a relative named Idanae Bing (granddaughter of a Nish Bing), please let me know. She shows up in my tree as a DNA cousin.
Hope that helps.