If the Lee scale seems to vary between various charges, I would also double-check that you have it--the razor edge--centered properly on each measurement. It is not hard to get the balance off a hair and have binding.
Then, as others here have heard me say before--the vernier slide is most easily used by
1) taking the beam off the balance,
2) leaving the vernier friction pin 'in'--keep it dragging.
3) positioning a thumb on either side of the vernier slide and
4) adjusting the vernier with a 'push' with your thumbnail from either side, as needed.
Because the friction pin is on, the slide will move positively--and you will soon learn the amount of push needed to accurately adjust the vernier.
When you replace the beam, of course, you do need to check the ball for the 10s setting you need, confirm the balance is clean, and that the scale operates smoothly.
As for the accuracy issues here--I vote with the balance beam being the accurate one. I now have had two (three?) cheaper digital scales, and those go south every time. Besides standard life-span / accuracy issues, these scales start to drift and change reading as the battery voltage goes down--e.g., as a battery discharges.
I use my digitals for tare weighing--and, I do own a own set of check weights.
Jim H.