The Ohio Court of Appeals has addressed a nearly identical situation in State v. Paseka.
The relevant law is, as you noted in the question, R.C. 4511.39, which states, in relevant part:
No person shall turn a vehicle or trackless trolley or move right or left upon a highway ... without giving an appropriate signal
The facts in this case are directly on point—an intersection where continuing straight puts one on a different road, while the original road requires a turn to stay on it:
Appellant was traveling west on State Route 6. At a certain point, Route 6 veers to the left. As appellant approached that area of State Route 6, he chose to maintain a straight-ahead course which automatically placed him on Wahl Road. He was stopped for failing to activate his turn signal in violation of R.C. 4511.39.
The court ruled that this did not violate the law requiring the use of a turn signal:
It is undisputed that appellant’s straight-ahead entrance onto Wahl Road did not require him to turn his vehicle, nor did it require him to switch into a different lane. As such, we fail to see how appellant violated R.C. 4511.39.
Here's the intersection in question, via Google Maps:
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