It's my second turn: my opponent controls a battlefield with an Otterpus; I attack, kill the Otterpus, conquer the battlefield. Do I draw or score?

You score 1 point. Otterpus's ability does not apply here because it's already dead before the conquer happens.

Here's why: During combat, damage is dealt in Step 2, then the Combat Cleanup in Step 3 kills all units with lethal damage (Rule 323.5 / 3b) — that's where Otterpus, a 2-Might unit that took 2 damage, goes to the trash. Only after that does the Combat Cleanup heal remaining units (3c) and optionally recall attackers (3d). Then control is established and you conquer (Rule 461.5.d).

Since Otterpus is in the trash when the conquer occurs, its static replacement effect ("draw 1 instead of scoring on turns 1–2") is no longer on the battlefield to apply. You gain the point normally.