I'd have to be in the Room, I was Not .
making sure the rod and barrel adjuster are as close together as possible?
obviously you are not mechanical and I am not a professional technical writer..
frankly I have no idea what that refers to,
the barrel adjuster is just the way you fine tune the cable pull length so the end of the shaft that the chain is riveted to,
can be brought into the proper adjustment..
I had no problems with my AW3 in 1957, and I removed the trigger shifter to use a friction lever.. ( I added 2 derailleur gears , for a 3 by 3 by 3 speeds) 27.
I think you need to read more about Sturmey Archer 3 speed hubs
https://www.google.com/search?source...54.Po5X6D2OGtU
....