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You could try a 12BA6 in place of the 6BJ6. pins 2 and 7 are swapped, but as one is the cathode and the other G3 (which is usually at ground) it should work. The tuned circuits around this tube would be a little off, but if you don't get any better results than what you currently have with a 12BA6, the odds are it's not the 6BJ6 tube and you can put it back in.
The 12BA6 and the 6BJ6 have the same heater current, and in a series string set that's what you want.
It's possible (and likely with wax caps still in the set) a B+ or screen grid bypass cap has gone shorted. That could drag down the set's B+ enough to disable the FM local oscillator, and make hum on AM. And you might as well replace the other wax caps while you have it on the bench, the rest of the wax caps will soon fail as well.
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