You are correct that the Doctor was just over 1100 years old at the end of Series 6. He may have still been aging during the alternate 5:02 PM timeline, but because River went through with the murder and time righted itself, that technically never happened, so he never got the opportunity to age like that. He's as old as he was before his "murder."
the Doctor wasn't "rapidly aging" during the TimeLockedMoment. Because that period of time was not progressing. It's possible that they were in that same second? for a very very long time. The whole point of the anachronisms was that time developed in that one time. Things started as they were then (more or less) and progressed. The Doctor became the soothsayer after years of speaking on the same thing 'Time is stuck', but because Time was stuck noone really aged. They just kept living over and over in that same moment.