Ringo only solely penned 2 songs in the Beatles, Don't Pass Me By and Octopus' Garden. He did that with George's help and also used George's help with songwriting during his solo years. Yellow Submarine and With A Little Help.. were written by Lennon-McCartney but mostly by Paul.
It is true that some of his parts on the White Album were re-recorded by Paul. But that was fairly common during those sessions. Those guys would come in separately and record their own tracks and work in different studios with different engineers on their own schedules. On the Anthology series where Paul George and Ringo are listening to a track in the studio making comments, one of them asks, who played bass or guitar on this? George remarks with a look of half disdain half laughing, "it was Paul, it was all Paul! Listen to it!". Something like that. Ringo wrote Octopus' Garden during the 2 weeks he quit during the White Album recordings until they begged him to come back.