I have 3/8 inch particle board which I am screwing and gluing down to 5/8 plywood. How close together should the screws be?
It will create a mess if the particle board is not predrilled because the screws will at least break-out the backside of the particle board there by preventing the screw from ever drawing the PB flat against the plywood ... especially if the PB is not high density (heads will pull through) ... and especially if the screws are going into something as thin as 5/8" (low grade) subfloor grade plywood.
When predrilling, a depth stop will be essential to keep from drilling into the plywood because over-drilling by a mere 1/8" will leave less than 1/2" of plywood for the screw to grip. A small chunk of wood can be used as a positive (failsafe) depth stop. It'd be best to layout the joist locations on the 3/8" PB and aim the screws to grab into the joists.
If adhesive from a caulking gun is used, the floor will end up wavy, not flat, if you were to put a straight edge to it afterwards.
Regardless, I can't see the job coming out well using 3/8" particle board. Sorry for the pessimism. Is 3/4" not an option?