QMy opponent's leader has 12 or less defense and I have 0 cards in my deck. If I have at least 12 cards with different base costs in my cemetery when I play this follower's On Evolve ability, do I lose the game because I can't draw a card from my deck before 12 damage is done to my opponent's leader?

ANo, the game would end in a draw.

While the "draw a card" part of the On Evolve ability is resolved before the 12 damage to the opponent's leader, the game loss doesn't resolve until the next confirmation timing. Since both you and your opponent would meet the conditions for a game loss at that point, the game would end in a draw.