Yes, the 57 twin does have a cathode follower acting as a buffer to drive the tone stack. It is the third tube from the right when facing the back of the amp. 12AX7s with spiral filaments cannot handle elevated voltage on the cathode; therefore, spiral filament tubes are not the best choice in the cathode follower position.
Maybe they're just bad tubes, as you say. However, they do seem to work OK in my other amps.
I'm just trying to find out if the problem is with the tubes, the amp, or a mismatch between the two. Strange, a recent Tonequest Report suggested Tung Sols for the Tweed Twin. I wonder if they actually tried them?