When you fill the last vacant hex on the board a battle begins immediately. You resolve the battle (which will inevitably leave many open spaces) and then continue placing tiles where you left off.
You can find that on page 3 of the rule book at the end of the "using tiles" section.
If at any time during the game the board is filled up and no single space is left to place a tile on, a Battle commences (see BATTLES section).
The Final Battle only begins when one player has drawn his last tile. He finishes his turn and then the other players finish their turns and then final battle begins.
This is also on page 3 in the "ending the game" section.
If any of the players draws his last tile from the deck, he can finish his turn. Then the opposing player finishes his and the Final Battle begins.
When any of the players has drawn his last tile, Battle tiles cannot be used.
If a player has used his last tile and filled in the last unoccupied hex on the board a Battle starts. Then, after the Battle the opposing player can use his remaining tiles and the Final Battle begins.
and in the multiplayer rules on page 8
- The Final Battle: If any of the players draws his last tile from the pile, the others complete their own turns as usual, and only then does the Final Battle start.