Bub being engaged doesn't mean you'll go into labour anytime soon. DD was engaged from 28 weeks and full engaged from about 37 weeks yet i still went a week overdue.

DS was engaged from 30 weeks and fully engaged from about 38 weeks. i went into labour 4 days early.

But still do take it easy when moving because any stress(even moving) can put you at risk of going into early labour.