Omega was incorporated in 1970. In 2010 the census showed a little over 1200 citizens. You don't usually have a police department until you are an incorporated municipality. So your badge cannot be older than 1970.
Although I know next to nothing about police hat badges, I would (for the moment, anyway) caution against the hard limit of 1970 for a date.
An "incorporated" town or city simply means that it has a charter from the state to have elected offices.
Here's a Rand McNally map from 1910 that shows the town of Omega, GA.
The map doesn't say, but presumably Omega wasn't incorporated then (and subsequently dissolved / only to be re-incorporated later in 1970).
Seems pretty unlikely.
GeorgiaInfo
So I guess the only question is, could you have an
unelected Police Chief prior to the town's incorporation?
Answer: I don't know.
Curiously, the 1910 Rand McNally map shows Omega, GA wholly contained in Tift County, but some other web resources claim Omega straddles Tift
AND Colquitt counties.
There's also an "Omega School District" described on an even earlier "Hudsins Co." map: 1905. (See lower right)
Old County Maps | TIFT COUNTY GEORGIA (GA) MAP BY THE HUDSINS CO. 1905
Terms like "School District" make me think "official office", but whether elected or unelected, again I don't know.
It could be that these official offices (if any) are administrative subdivisions derived by statute, and not by state charter (i.e., incorporation). [From Wikipedia].
And if so, then the date of Omega's incorporation may have little bearing on the age of the police chief hat badge.
Though, I wouldn't rule it out.