Is anyone running a 4425S Carter on a 68 340 engine?

Yup. Look at the exact service manual for the carb. Also check the TSBs (at Hamtramk Historical).
Many of the CAP carbs got a more limited range of idle mix adjustment. The single screw is almost certainly an idle air bleed adjustment screw. Some of the two adjustable screw carbs are also into the air bleed passages.
To meet idle emmissions these carbs were meant to be finely adjusted using a CO sniffer. The goal was a relatively lean idle mix, around 14.2 AFR by '69.

Mymopar might have a better scan
1968 Chrysler Imperial Carburetor Service And Repair Highlights from the Master Technicians Service Conference Session 243