Since it seems nobody else is willing to hazard a guess on your brass 3-piece Savannah Police button's age, I'll try. It appears to have what civil war button collectors call a "wide rim" (in comparison to the typical civil war period "narrow rim" on 3-piece buttons). The wide rim became popular a decade or so after the civil war, approximately 1870s/80s. There's even an "extra-wide" rim version. Because the words narrow, wide, and extra-wide are open to interpretation, here's a photo (below) showing New York State Militia 3-piece buttons with (from left to right) a wide, narrow, and extra-wide rim.
All of that being said... your photos of the still-encrusted button aren't quite clear enough for me to be 100%-certain that it has a wide rim. With the photo below, you can judge your button's rim size better than me.
Also, please try cleaning up the button's back and tell us any letters you can discern in the button's backmark. Well-focused closeup photos of the cleaned back might also help.