The wiring in your diagram is correct.
Your disconnect problem sounds like an electrical noise issue. If
it seems to run OK without the 36 power supply turned on (but of
course the motors will not move) then it is likely noise.
Check to be sure that the power wiring to the power supply, from
the power supply to the drivers, and the wires to the motors are
not directly beside the USB cable or any of your home/limit switch
wiring.
If your computer is a desktop, try connecting an 18 AWG ground
wire from the computer chassis to the safety ground in your
controller box.