Drywall screws snap off when trying to fix squeaky subfloor

By "drywall screws", I think you are meaning 1 5/8" blued screws. I do not think that they would be a good choice. We usually shoot down plywood subflooring with 2 1/2" to 3" ring shank nails, however 2" to 2 1/2" decking screws work well, but just a lot more work. I would encourage you to use a little adhesive on top of the floor joists or strips of felt paper or Grace ice/water shield to avoid any squeaks later on.

I would reset all the nails that are holding the 1X6s,making sure the nail point has found new wood by setting the nail head below the surface by making the last few strikes with the hammer at an angle to dimple the wood with the nail head. With the nails drawn tight, that should eliminate any potential floor squeaks, but for redundancy add a screw between all the nails. Do not add no more than that, the potential of splitting the joist tops. So if you can, run the screws in at an angle to get the screw to not go the same direction as the other nails. It is a minor detail, but that is what I would do.

After your existing subfloor is secured do what you can to get your T&G joints to break near the center of the 1X6, running you plywood the same direction as the 1X6, perpendicular to the joists.

In my opinion, 3/4" along with the existing 1X6, is overkill on the subfloor/underlayment combination, unless you are setting tile. 5/8" T&G will be a lot easier to carry upstairs and still give a T&G feature which is good. In a minimalistic approach 1/2" underlayment grade plywood will work very well too with the re-secured 1X6. DO make sure it breaks on the center of the 1X6s, cut it if it is not, I refer to NON T&G plywood here so ripping an inch or two off the occasional piece does not matter. What matters is it breaks on the 1X6 everywhere. After that 1 1/4' OR 1 5/8" screws on a 8" grid pattern and construction adhesive will tie it all together. There is no need for 3 1/2" screws for any reason. Maybe 2 1/2" in the joists here and there if you choose, but remember the thing mentioned about too many fasteners in the tops of the joists.