To me, there was no one period where John or Paul "peaked". From their earlier stuff to the very end, they always sounded great to me. To say that John's vocal on gems like "I Should Have Known Better" or "Twist and Shout" and Paul's vocal on "I'm Down" or "Yesterday" were inferior to anything done after is a disservice to their immense talents.
As far as getting chills, I have to go with Lennon's vocal performance on A Day In The Life; in terms of expression and voice control (in conjunction with the echo effect), I feel this track is his finest moment as a Beatle vocalist..so he definitely had a ways to go beyond the Rubber Soul period.
For sheer energy (Happiness Is A Warm Gun, Yer Blues)combined with that emotional tenderness he often displayed (Julia), he excells on The White Album which, I feel, is his peak moment, overall, as a vocalist. This isn't to say he wasn't good on various levels afterwards, but he never surpassed this period (add to it vocals from songs such as Hey Bulldog and Revolution from the same year).
Paul did not peak within the Beatle years. He basically kept getting better and better until the mid-seventies. The very first indication of a strained rasp in his voice is on the title track of Venus and Mars..and, although he still continued to have one of the finest voices in Rock, the rasp became more and more evident in the live performances (listen to any live performance of Coming Up or Maybe I'm Amazed) and it's quite present throughout Back To The Egg.
That said, he had quite a few "good years" ahead of him both in terms of the studio and live performances but the very best were behind him.