You have two issues here.
Firstly, make sure both the headstock register (the bit after the thread next to the headstock (sorry if that is teaching & eggs)) is 100% clean.
Secondly, using an adapter will lay you open to more vibration. The further you are from the supporting bearing, the higher the dynamic moment forces are (10x static loads often) and you will see a bigger effect.
So, what can be done. Simply putting a SP chuck on will not help unless you use it without the thread adapter but it is really a big, heavy chuck for the machine which (with all due respect to perform) will not have top of the line, grade A1 industrial bearing fitted in the headstock. Is the thread in your existing chuck changeable? If so, I would go that way.
Also, have another look at your set-up. It the lathe well held in place? You could either bolt the feet down (put a rubber cushion in-between the floor and the legs if bolting to concrete) or build a sand-box under the shelf in-between the legs to add extra, vibration absorbing, mass. (Really dry sand is the best for this and will be much more effective that wet sand).
If you are still stuck, as I said on the other thread, give either Colwyn or Jason at the APTC shop a bell and they will be able to sort you out I am sure.
BRgds
Simon