You can double-check the jet centering by turning the mixture nut as lean as it will go, then lifting and releasing the piston. It should land with a distinct click, as it does when the jet is lowered. If you don't hear the click or it is much softer, then the jet is not centered.
I found the tool to be pretty much useless; in addition to not making any sense. If the bore in the carb body were always perfectly on-center to the bore in the chamber (as the tool assumes), then they could have just made the upper gland be a snug fit in the body and not need centering at all. But when my jets are properly centered to the needle (as witnessed by the drop test), they are visibly off-center to the bore in the body. Not by much, maybe only .003" or so, but some.
Like Charley, I center mine by trial and error. Tighten the nut until the housing will just barely move with gentle tapping, then tap it around until the needle moves freely. Finish tightening and hope it doesn't move
But, I doubt that is the problem with the jet moving. Try polishing the surface of the jet until it shines (to reduce friction with the seals). I found that ordinary toothpaste & my fingers would put a nice shine on them. (Obviously clean thoroughly afterwards.) Might also be worth replacing the springs, especially if you don't know their history. They can lose tension with time and corrosion; or a DPO may have tried to stretch (or shorten) them to "fix" a problem.
PS, I have also found that there is enough play in the joint between the dome & body to make a difference in centering. If you remove the dome and reinstall haphazardly, it may be off enough to cause binding, which can affect idle mixture and lead to wear in the jet over time. So, when I install the dome, I put in all three screws finger-tight and then turn/hold the dome clockwise so the screws help locate it for the final tightening. That seems to be more repeatable, enough that I can R&R the dome without having to re-center the jet.