Did I use screws that were too short?
I'd personally be a little concerned they were a tad short, mainly if they were the sole means of fixing (but also in a different context1). As I mentioned in the Comments, if you glued as well as screwed then the glue will of course be doing a lot to hold this cabinet together, and in an ideal world could be used by itself with no brads, nails or screws to help lock everything together.
did I pre-drill it with too big a bit?
Sorta. But if the combination bit was adjusted exactly as seen in relation to the screw, no I don't think so. You certainly would not want to drill a full-length pilot hole with a bit of that size2 however!
Note: a proper combination bit for pre-drilling in a single operation should either have a longer, tapered bit in its centre, or be stepped, so that it drills two distinct holes sizes — for the clearance and pilot portions of the hole.
1 Given this is not a freestanding cabinet and will be built into the wall though, I think you have zero to worry about.
2 Remember there are two or three components to pre-drilled holes, the countersink/counterbore on top (IF present), the clearance hole that the screw should not engage with (assuming the threads extend up that far), and the pilot hole. The pilot hole should be approximately the core diameter of the screw used, no larger, and in softer wood and plywood (also MDF and chipboard) you can get away with under-drilling the pilot to maximise hold.