Will godaddy still be in my email loop if I only move email to another host?

(I know this is something I can research online, but I won't have time to do so by the time I need to make decisions.)

My company wants to get rid of godaddy as their email host company because their servers in Arizona keeps intermittently rejecting incoming emails from one of their customers, citing "[IP x] has sent to too many recipients this hour. IB607".  The sender customer is a government agency. WE called godaddy support and the guy said, "I don't have a resource to say, turn off the filters for messages coming from this server or that one."

So my company plans to move to a new email host, but MIGHT keep the company website and/or domain registration with godaddy, with whom they have a multiyear contract.

My newbie question (I need to be sure):  If they move their email to google apps (lets say) but keep registration and webpage with goaddy, WILL their email traffic still be passing through godaddy's servers? Will it simply be forwarded thru godaddy?  (In which case this would not solve the problem.)

Thanks