Just to let you know, the fireworks do not start until Tues. June 19th this year.
Also to let you know Disney is in the "Shelter Cove" part of the Palmetto Dunes Plantation. But I think you are talking about the marina and yes it is a very easy walk. There are a bunch of restaurants and cute shops there.
There are a bunch more shops in the Coligny Beach area too which is more south on the island before you get to Sea Pines.
Personally we have such a good time in HHI we don't leave the island. However we are there later June and live on the beach a good portion of the time and also do many of the activities. But if you are not so much into the beach or its chilly a drive into Savannah might be fun.