Don't Replace, ADD
Removing the screws is a lot of extra work, for no real benefit. In fact, you might even find that some of the old drywall screws snap when you try to unscrew them, and then you have no practical choice but to add other screws anyway, leaving the partial screws in the studs.
Just add proper cabinet screws. Depending on the layout, you should be able to move a little up or down or left or right from each existing screw and put in a cabinet screw.
Predrilling holes slightly smaller than the new screws is highly recommended. That will make it easier to drive the screws, minimize the risk of any cracking, and also let you easily tell if you are hitting a stud (which you want here).
Thinking about this a little more, this is definitely a case where having two charged up drill/drivers would make the job much easier - one with a drill bit, the other with a screwdriver bit. If you've only got one, I'd pre-drill a bunch of holes then come back around with the screws.