Although there are 5 terminals some do not have car parks, and they provide a shuttle. ABP seem to have limited space and ask for your cruise Co and departure date, it is not uncommon for there to be no availability.
Have done 2 cruises from Southampton this year, one coincided with a rail strike, we managed to get parking just before it ran out, but it was about £90 for 5 night cruise, the other much longer cruise we took train from Wiltshire (free parking near station in safe area). Cheap GWR group save ticket to Southampton.
At Southampton it's about £8 for taxi, which we rounded up to £10 cash. Lots of taxis shuttling from station to the cruise terminals, if very busy might need to wait couple of minutes.
Haven't used off site parking.
One other tip, they might give you a check in time, but in practice that seems to be to stagger arrivals, anything between about 1 and 3pm, even if 90min early they let you check in and board.
For 2 week cruise might be worth getting train to Southampton, as parking at short notice will be steep (if you find any)