My suggestion to fix the Input problem:
Let's say you want to move like 1-2-4. If all 3 Inputs are pressed in quick succession, I'd only store the 2, check if there is a stored Input and move the die accordingly after the animation ends. The 4 is lost to avoid the player from chaining to many Inputs. You'd have to be careful not to accidentally store the 1, but that should be a pretty simple fix.
Happy to hear about the post jam update, I'll be looking forward to it!