Ive done 2 Disney cruises from Southampton and a 3rd one booked for August.
Disney is using Horizon Terminal. There are 3 hotels less than 10 minutes walk from this terminal, IBIS Southampton Central, IBIS Budget and Novatel. All 3 share the same carpark and are located next to McDonalds and TGI's and across from the main shopping centre West Quay. Ive walked to the cruise terminal with my luggage ad it was an easy walk.
Holiday Inn recommended by your TA is a budget hotel. Its a bit further walk from Horizon Terminal about 15 minutes walk. The walk is easy as its the same side of the street as the entrance to the cruise port.
Leonardo is a mid price hotel. Its a bit further than the others from Horizon terminal and would be a 20 minute walk, as you have to cross the busy street at the cross walks.
Alot of UK people stay at Premier Inn, which is near Leonardo. Its also a mid price hotel.
Cruise embarkation mornings are very busy in Southampton, there can be up to 6 ships embarking and debarking in the morning. The walk to the cruise terminal from all the hotels is on the same road, they are just varying distances from Horizon terminal.