Hit this on my M3 Max MBP on 26.5 — it's not fully broken, the drag registration just got stricter. What worked for me: click and hold for a full half-second before you start moving, wait for the icon to do a tiny "lift" scale animation, then drag. If you do the old instant click-and-drag in one motion the Dock treats it as a plain click and launches the app instead. The tell is whether you see that little scale-up before the icon starts following the cursor.
If even the hold-then-drag isn't taking, `killall Dock` in Terminal usually clears it. I've had to do that twice on the MBP after wake-from-sleep specifically — Dock daemon gets into a state where drag events don't register at all until it restarts.