Cutting insert cartridge for chip removing machining tools has an axial adjustment system threadedly engaged with a cartridge body and a tool body. The axial adjustment system includes a differential screw in threaded engagement with the tool body either directly, for example by threaded engagement with a threaded opening in the tool body, or indirectly, for example by threaded engagement with a cross dowel fastener slidably positioned in an opening in the tool body. Actuation portion positioned axially between threaded end portions of the differential screw has an engagement feature on an outer surface to be driven, e.g., with a tool or by hand, to rotate the differential screw and adjust the axial position of the cartridge. The spaced apart relationship improves access to the actuation portion and engagement feature for rotation of the differential screw. Chip removing machining tools with such cartridges and methods of machining are also disclosed.