After I beat the Elite-4, I can't seem to find Drilbut in any dust cloud, I only find Exadrill. is Drilbur gone forever?
Just check these areas:
You should be able to find Drilbur in those areas. Other than that, it is all Excadrill.
No, you can catch it in Twist Mountain,etc.
Post-game, wild Drilbur does not appear in new unlocked areas, only previously explored ones.
Source : This Site, Personal Experience.
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