Not very clued up on Southampton I'm afraid, but Pompey has a few good things to do, if you don’t like shopping.
As you say the historic dockyard is very interesting, but a bit expensive.
Other museums of note are:-
Fort Nelson Royal Armaments on Portsdown hill.
HMS dolphin submarine museum in Gosport.
And on the seafront in Southsea(Southern part of Portsea Island)
Royal Marines Museum.
Just a few miles down the coast to the East past nice places like Bosham is Chichester, with it’s shops and walks around the Cathedral and grounds, and to the West is the new Forest with the National Motor museum at Beaulieu. The Forest is obviously great for walks etc.
Of course there are lots of other ‘boaty’ related places to visit, and lunch at The Rising Sun in Warsash is worth looking at.
RIBBED For extra pleasure.
Member of the ebay Blue RIB cover club.
Member of the Bombard 380 Aerotec club