Does the Double Tree Hilton have shuttle service from the airport to the hotel and to the cruise ship port?

Yes. It might be a part of a stay and cruise package. The shuttle only transport in a short radius.

I know I walk bye there sometimes but I do see a shuttle

There's a complimentary service to the cruise port but you need to book your time when you check in. From the airport to the hotel I used an airport shuttle service (cheapest option)