Good morning, there is not 1 answer to your question. Some people like the 17th Street area which has hotels and restaurants with some shops as well.
Other people (I am in this group) stay on Las Olas Blvd......there is a specific hotel there that I stay at call The Riverside and there are lots of shops and restaurants along the Blvd.
Some folks prefer to stay beachside.
Is this your first time sailing or staying out of Ft. Lauderdale? There are lots of threads on here about the area.
As for Sawgrass Mills? That is very west of the port..Yes, there are lots of stores and it is a big mall but it is not what I consider "close by". You could not walk there, that is for sure. Uber, Taxi, Limo, some kind of transportation is definitely needed to get there. It is probably between a 20-30 minute car ride from the hotels that are near the port.