Any chance it could be MTU related? Sounds like the issue is only popping up when attempting to use full packets.
For me, it's always DNS until proven otherwise. But the difference of some sites loading, but others not makes me suspect there's a split somewhere and IPv4 vs 6 seems as likely as anything.
It has caused some significant weirdness, and I've had to relax the rules just to get certain things to work at all.