It shouldnt make any difference what sound tube you put in there. Most tubes end up being marked as both the 6J10/6Z10. Your sound problem is most likely and out of adjustment quadrature coil. Ive run into this a few times. the quad coil on your chassis is DIRECTLY to the right of the 6J10, its a coil mounted close to the chassis. The other two coils to adjust are the intercarrier coil (First coil directly to the left of the 6J10) and the sound take off, which is to the left of the intercarrier coil. Both of these coils are in the aluminum cans. And the buzz control of course. Those are your 4 adjustments that will affect your sound and output quality.
Yes, it is best to tune to a weak station and attenuate the signal. If you dont have the attenuator then just short the two screw jacks together with an incomming signal and try to tune in the worst hard to receive station. For me its CBS 2. In fact, I dont even need an attenuator if I adjust to that station
. In order to make this adjustment as best as possible, you want to hear the "hiss" in the background of the sound. What you want to do is alternately adjust all 4 of these controls untill you have acheived the best possible clearest and loudest sound. Most likely though, adjusting the quad coil and the buzz control alternatly should do it, however, if someone was in there tinkering, then you will want to adjust the other two. Obviously make sure you mark your adjustment tool so you know how many times you've turned a coil. This should do it. Ive had a number of sets in the past, and one of them was a 24MC32 with bad sound. In my case, I just turned the quad coil and the sound exploaded with power. Its not uncommon to need to "touch up" the quad coil in a zenith set.
As for your jittering vertical, you probably have a dirty set up switch. Time and time again Ive had jittering vertical, all caused by a dirty set up switch. Just switch it back and fourth a few times should stop that trouble. It will probably come back though. Your best bet is to shoot some tuner cleaner in that switch and work the switch a few times.
These have always been the cases for me with troubles like you describe. Hopefully its nothing different for you. Good luck.