Part of the educational experience gained here... I picked up a really good deal on a fairly new '68 Princeton Reverb RI that had exactly the same issue, plus a cabinet rattle. It's similar to my '65, but a totally different voiced combo-amp. The '65 is a15w and 12" speaker, while the '68 is a 12w, 10" speaker, but same general configuration. This one had the same weird harmonic distortion on certain notes. I seriously think Fender might've got a wonky batch of GrooveTubes 6V6's.
Longer story shortened; guy was tired of fiddling with it, and even more frustrated than I had been with mine. (less patience, which is hard to imagine...

) Anyway, Jimmy J's screwdriver tap-test confirmed one of the power tubes was suspect, and new matched pair of J/J's cured that weird harmonic warbl-ey racket. It also had a rattle in the cab itself on the lowest A notes. I snugged all the screws, put some small loops of gaffers tape behind the backboard, and that seems to have cured it. I have read where the front baffle boards in these (particularly the '68 RI's) are prone to rattle. Mine seems to be fixed though.
I like things about each of these Fender Reissues but I really don't need both the '65 & '68... two nearly identical combo amps... I may eventually have to decide which one to keep. I have this fuzzy feeling there is an F-2B in my not-too-distant future. I can envision a trade happening.