This is the urban legend of the time (late 60's/ early 70's) about the end of CCR. I don't know if there is any truth here or not! I am younger than the band members and likely older than your parents, so I was a bit too young in the 60's to do more than just notice the music as played on radio (have you heard Tommy James & The Shondells - very cool!!!not kidding). Anyway, John Fogerty, finally tired of being ripped off by his record label, decided to cease generating money for them and start making money for himself - so he went on the road with a HIRED band - the kind you pay wages to, not share the royalties with. Seems like I've heard of a law suit, but I'm fuzzy on that one - Google the newsgroups or locate historical archives - I don't have time right now because I'm copying their 60's vinyl (the original purple and silver Fantasy Records label) - into WAV.CDA.MP3.WMA files.
John Fogerty was a one-man-hit-writing-machine (he wrote most of their top 10 hits) and therefore wrote most of CCR's original music. In the 1960's, generally the record publisher took the electronic reproduction rights and the song writer retained the public performance rights. So yes JF can play his own songs in a concert or live media production - and he will retain his part of the contract money for performing. Look at how Paul McCartney has profited from his part in the Lennon/McCartney songs of The Beatles since their breakup.
Your research project is to compare how Fogerty and McCartney managed their intellectual resource base (their songs) for greatest income over time. They chose very different methods and I think that McCa did the best. Maybe he was the better artist or did he have better help managing money? Ask and ye shall recieve - look and ye shall find!!!