In English, "will" often means "am willing to" (usually reduced to a contraction).
As, "I will help with the dishes." or more commonly "I'll help with the dishes." I would probably only say the former for clarity in a noisy room.
I have no sense of the Dutch. "Ik wil met de afwas helpen." might mean that it's something I really want to do.
If I have understood right "zal" would be a bit (perhaps comedically) overdramatic when sitting around the table wondering who will clean up.