Will dual-channel memory improve my 5600G's gaming performance?

Yes. Adding a second 8GB stick to run dual-channel is the single biggest performance upgrade you can make short of buying a dedicated GPU. Your integrated graphics uses system RAM, and single-channel chokes memory bandwidth to roughly half its potential — expect 30-50% higher frame rates in many games. Get the exact same model if possible, but any DDR4-3200 CL16 stick will work.