Well, the status of the crew in ST5:TFF is more or less explicitly explained anyway: they were supposed to be part of the crew of the E-A, but that ship suddenly and unexpectedly started misbehaving, giving everybody but Scotty a surprise holiday of a few weeks.
Between ST5 and ST6, though, I'm sure the older officers did semi-retire, while the E-A continued her adventures under other commanders. But Spock, despite finding time to do top-level Klingon diplomacy, apparently remained aboard the E-A in the role of Science Officer and/or perhaps First Officer, a role he wanted Valeris to inherit.
I wonder who was Spock's CO during those years? In theory, it might have been Kirk: he was among those who didn't blather about buying boats or hosting conferences.
As for the E-A having a short career, I think that was solely because of the ending of the long cold Klingon war. Kirk was certain the ship had plenty of life left at the conclusion of ST6 (indicating it was the officers who would get decommissioned, not the ship). He was simply sorely mistaken; Starfleet probably wanted to retire the Infamous Ship of Kirk ASAP to make the Klingons happy.
For all we know, the E-A was originally launched at the same time the E-nil was, and just refitted a bit differently, or perhaps a few extra times. She might have been in much better condition than the E-nil due to the extra refits, but other reasons dictated her decommissioning, and the launching of a new Enterprise
untainted by the name of Kirk.