This is game and situation specific. So, I went and consulted the relevant areas of Robert's Rules of Order. Accepting or rejecting the card is determined by the game. But, that doesn't seem to matter here.
You have to accept the card, unless it was exposed due to dealer error, which does not appear to be the case from your version. You saw it because the dealer was careless and you were paying attention. It was exposed by your own actions and behavior, not the dealer's.
It is strictly against the rules for the player being able to choose whether they take the card or reject it. The required action is determined by the specific game, and for draw poker (high) you aren't allowed to take it; but, that is for a card exposed by dealer error. Like I said, you revealed the card to the table. That doesn't mean you get to reject it.
Get a cut card. Practice better procedures for shuffling and cutting. That will prevent this from happening again. But, the right choice appears to be made.