I agree with others that the check is the character's action for that round, and you get just one check against the whirlpool per round.
I disagree with your DM about Dash though. A character with Cunning Action (CA) can Move their base speed and then perform an action, like Attack. If their attack drops their opponent, they can then decide (after already moving some) to use their CA to Dash to another opponent. It can't be affecting their movement from the first part of their turn (making them faster) at this point. One
could always hand-wave it away, retroactively saying that the CA Dash
was chosen at the start of the turn, but it flies in the face of the flow of events, and could rupture the space-time continuum.