[XSleeper]

Harvey,

It's not so much that you "shouldn't" screw into the joists as it is that you shouldn't JUST screw into the joists... you want to put more screws in the subfloor that are closer together than the joists are. Every 8 or 12" spacing, for example. You want to fasten the 2 layers of subfloor together so that they act like one thick layer. 1 5/8" deck screws would be all you need.

A 2nd layer of subfloor doesn't need to be screwed into the joists for any particular reason. The screws don't need to be long enough to go into the joists either. There is no benefit of doing that.

 

[XSleeper]

If you have the ability to add tight fitting solid 2x6 blocking between those joists at midspan, that will help a little. Snap a chalk line and put one piece on one side of the line and the next piece on the other side of the line. This allows you to nail them all from the back side. If it needs to be rock solid, you'd need to put a header under the floor near midspan.