Why can’t we use déménage’ instead of emménagé as both mean to move as in to shift.

In this context, déménager means to move out/relocate, and emménager specifically means to move in(to). Having looked at the scenario, the interpretation is probably that 'she moved there' means 'she moved in'; if it said "she was moving there" (present/future) or "moved from" (somewhere else), I think déménager would be correct. In presenting the scenario to my wife (French) she used emménager in this context, and would not have used déménager as an alternative.