this is a common problem , slow down your z axis speeds will help enormously , your simply running at higher rates than the stepper motor can cope with , ideally you dont want a 10 microstep driver on the z .
as a start i'd lower them by 50% , or untill you have them running fine and then increase slowly untill they fail and then drop back about 10%
the faster a stepper goes the less torque it has
( and why you find z axis tend to have a 2:1 reduction pulley incorporated )
running 4 steppers could be dragging the system down , i'd tend to run the Z axis on a seperate supply ,
if the machine is not ridgid , then thats the first port of call to fix, electronics can only do so much , it cant make a machine better than it's frame can support
dont forget a stepper uses more current when holding a stationary position ,