If I react to Drag Under with Tideturner to move my unit to the other battlefield, does the unit still die?

Yes, the unit will still die.

According to RiftJudge FAQ #7179, when a spell like Drag Under targets a unit at a battlefield, the targeting requirement is simply "a unit at a battlefield." The spell does not lock the unit into staying at that specific battlefield.

Here is the timing sequence:

1. Original State: You play Drag Under, targeting your unit at Battlefield 1. The spell is added to the chain.
2. Reaction: You react to the Drag Under by playing Tideturner from hidden.
3. Tideturner Resolves: Tideturner enters play, and its triggered ability resolves, moving your unit from Battlefield 1 to Battlefield 2.
4. Drag Under Resolves: When Drag Under resolves, it checks its target. Because your unit is still at a battlefield (Battlefield 2), it remains a legal target for Drag Under's instruction to "Kill a unit at a battlefield."

The unit is killed as intended, regardless of the fact that it moved battlefields in response.