So you have two problems, a shifter that moves side-to-side when you first start out, and a vibration from the drivetrain you feel up through the seat of your pants?
If the shifter moves side-to-side as you get on/let up on the gas pedal, that is often caused by a broken motor mount from the engine lifting up off and then setting back down onto the broken motor mount.
Yes you can install a CV driveshaft to cure the drivetrain vibration caused by the 4" suspension lift, but you'll also need a SYE kit and adjustable upper control arms to get that CV driveshaft working properly.
Instead, you could install a set of 1" taller motor mounts like from Brown Dog (use his rubber, not poly mounts) which works like the t-case drop does. Installing the 1" taller motor mounts would be like adding another inch to the t-case drop. That would probably be enough to stop the vibes from a 4" lift. No guarantee but probably. If not quite, you can always drop the t-case another half-inch with washers. Of course the SYE/CV-Driveshaft/Upper Rear adjustable length control arms is the right way to cure those vibes which would also allow you to get rid of your present 1" t-case drop.