Sign conventions for g:

Kinematics: We usually call the direciton of gravity the -Y direction, so g is negative

Force problems: We usually assign the signs of forces (+/-) by inspection, so g shows up simply as the magnitude, 9.8.

Centripetal forces: The direction of the acceleration is constantly changing, so really only the magnitude of g makes sense where it shows up, usually in some kind of normal force, mg.