Should I overlay or replace 3/8 inch drywall during my room update?
I would just take the 3/8" down and install 1/2". Here's why I suggest it


Opening the walls would let you fix things if need be. Want to add some new outlets? Remove older wiring? Add some coaxial or Ethernet? Now's your chance. You said you were adding windows. You'll have an easier time if you just take the old stuff down.
1/4" is less commonly used, so it is slightly more expensive than 1/2" (plus you really need to glue the new boards to the old and that starts adding up). You could always stick 1/2" up over the top of the existing, but if you're going through that trouble, demo isn't that much more on top of that
Speaking from experience, hanging new drywall on bare studs is easier than going over the top of existing drywall. There's no guesswork at studs.


As to how to attach, I tend to use both nails and screws. When you're mounting the boards, nothing is faster than nails. A few nails will hold the board up. Then I go back and add screws. Be aware that screws are easier to goof up, but I like the lack of movement later. Be sure to smash the nails into the board (as in dent the board around the head) or popping will drive you up the wall later.