Galley sink check valve...
Unless your 28.5 is plumbed differently, the galley sink should have a seacock that is controlled by the thru hull located in the cabinet beneath the stove. The ice box drains into the forward compartment of the bilge, just aft of the compression post. I don't see how you could connect the ice box drain to the galley sink.
As a matter of habit, our seacock for the sink is always closed unless the sink is in use, then it's opened, to be closed when the galley clean up is completed.
To avoid the moisture and the bilge pump running from melted water from the ice-box, I've started bringing 2 or 3 frozen 1/2 gallon jugs of ice to cool the ice box. If ice is needed, it gets stocked into the auxiliary cooler that's strapped to the seat above the toilet. Less water inside = less moisture = a happy interior

. Once the ice has melted, I have 1.5 gallons of water to use elsewhere, as needed.