If I react to Wages of Pain with Demon Shade and move my unit to the other battlefield, does the damage still go through?

Yes — the damage still goes through. Moving battlefield-to-battlefield doesn't break the target.

Wages of Pain only requires "a unit at a battlefield." Demon Shade resolves first off the chain and drops your unit at the other (open) battlefield, which is still a battlefield, so the target is still legal when Wages resolves and it takes the 3. The Gold token gets played either way, since that instruction is unrelated to the target.

The distinction matters only for moves that leave the board-zone targeting pool — moving a unit to Base (or bouncing/killing it) is what blanks the damage (Rule 359.3.e.5; the target must still be a unit at a battlefield per 355.9.b).