Could be that the axles are not engaging the inner splines of the side gears ?

It's a 657 chuck which is the pre 741 , does it have the early clutch type suregrip in it from a rear with tapered axles ? Which may require a longer axle ???? I remember john K. posting something about the early rears having a different thrust width ????

If the axle isn't engaging the inner splines then the problem the OP is having can happen , doesn't necessarily mean the chuck is bad. A change of the side gear and the correct thrust spacing would fix it ?