You are mistaken. The last dedicated training carrier was the Forrestal, which replaced the Lexington and as the link I provided earlier confirmed Kennedy deployed for both OIF and OEF in 2004. Was she neglected towards the end of her career? Yes. Did she continue to serve as an active fleet carrier up to and including her last deployment? Yes.
Upon returning from the Med, JFK entered the Philadelphia Naval shipyard for a two-year comprehensive overhaul. JFK completed this overhaul on September 13, 1995 and began her journey to her new homeport at Naval Station Mayport in Mayport, FL. After that overhaul, JFK became the new Reserve Operational Carrier on September 30, 1995. In November 1995, the first reserve squadron, VFA-204, came aboard for a short cruise off the coast of Florida.
Technically JFK replaced Forrestal as the trainer although the hull designation did not change. Forrestal was decommed in 1993 according to a USN web site. That also fits the timeline I posted. As far as I know the Navy now does not have a designated trainer and JFK was one at least part time 1995-2000 according to what I had read in Navy sites. JFK was the last as near as I can tell.