If I get the final point by winning a showdown or do I have to double conquer. The other battlefield is uncontrolled?

To win the game by scoring the Final Point, you do not necessarily need to "double conquer" (conquer both battlefields in the same turn). You simply need to score on both battlefields during your turn.

According to Rule 448.1.b:

• If you score through Hold: If you are at the point total required to win (1 point away from the Victory Score) and you score a battlefield by holding it at the start of your turn, you win the Final Point immediately.
• If you score through Conquer: If you are 1 point away from the Victory Score and you score a battlefield by conquering it, you only win the Final Point if you have scored every battlefield (through either Hold or Conquer) during that same turn.

Summary of how to win:

1. Holding counts as scoring: If you hold one battlefield at the start of your turn (scoring your 7th point) and then conquer the other battlefield during that same turn, you have scored both battlefields on your turn. This satisfies the requirement to win the game.
2. Conquering both: If you have not scored either battlefield yet, you must conquer both in the same turn to win the Final Point.
3. Uncontrolled battlefields: If a battlefield is uncontrolled, moving into it and establishing control counts as a "Conquer" (Rule 444.2.b).

In short, as long as you manage to "score" (either by holding at the start of your turn or by conquering) on every battlefield available in the game during your turn, you will trigger the win condition for the Final Point.