Yes, you have a cap badge. And a wonderful family treasure. Here is a close-up of a pair of carriers wearing theirs. It is cropped from a large portrait of 7 postal workers, I believe your cap badge to be somewhat older than 1920, perhaps c1905.

Also, here is a listing of the Hamersville Ohio PO employees in 1911, from p410 of the "Official Register of the United States 1911, Vol II, The Postal Service" Unfortunately, 1911 is the last year the PO employees were included in the Register. Finding later employee lists is more difficult.
