Sorry Caymaniac, don't know much about the "Diversity"... only liveaboard I've been on up north is "Aquarius III". I'm sure it'll be incredible though!
Lots of lovely beaches in Port Douglas - "Four Mile Beach" is just idyllic! And if you're there in June, its safe jellyfish-wise (they have the jellyfish nets out from about November to April/May).
As for the Yongala - I don't think there are any boats that leave Cairns, they pretty much all depart from Townsville, and it is approx 5 hours from Townsville so it would be MUCH longer from Cairns, so its probably not worth the boat's petrol money!!
It would be a real shame if you were to come all the way to Nth Qld and then not do it, but as you said, 5 hours is a long drive from Cairns to Townsville... However, there are loads of things you could do along the way - maybe stay a nite at Hinchinbrook Island, stop off at Mission Beach, lots of lovely coastline to enjoy! But yes, it is a long drive.
Regardless, you'll have such an incredible time in the coral sea, enjoy yourself and "best fishes"!
Cheers