The RR states
If Dracula plays "Escape as Mist" or "Escape as Bat" before he has played a number of combat cards greater than the number of despair tokens the card is canceled
So in the first week, Dracula can escape provided he has played one card.
The question I have is, does the actual escape card itself count?
We are assuming the card itself does not count, so Dracula must play at least one other combat card before he can escape.
Is that right?
The escape card does not count. Essentially the round when Dracula can first successfully resolve escape is Despair +
12, so in week 1 he must wait until the second round.
See the combat example in the Learn to Play guide which has a situation in the second week and it is stated that Dracula must wait until round 3.