Washer overflow drain is emitting sewer smells. There is no recent evidence of any water in the overflow pan of recent vintage, although there is "rust" in the pan from water at some previous time. Is it possible that the water in the P trap has evaporated and that the sewer smell comes in that way because there is no water to stop it? Or is the smell most likely coming from the build up the bacteria in the drain walls

Once you have water going through the P trap, vapors will be blocked.
Run water through any sinks, showers, or tubs as well.