Q: If I play a New Rule or Keeper that’s not a Draw or Play, but increases the Draw or Play, do I do that immediately?

A: Rules that increase your Draw or Play based on a formula (Doom, or what Keepers or Creepers are in play in front of you or elsewhere) as well as Keepers that change the way the rules work for you personally, should be treated as you would an increase in the actual Draw or Play Rules, so the total number of draws or plays you get on your turn increases:

If you had Draw 1 and played a card that made it Draw 1 (+1), then it’s just like you have gone from Draw 1 to Draw 2, and you draw 1.
If you had Draw 4 and played a card that made it Draw 4 (+1), then it’s just like you have gone from Draw 4 to Draw 5, and you draw 1.

Likewise, if such a card is already in play, increasing the Draw rule results in the exact same draw increase as though the formula were not in place.
See If there’s a Rule (or Keeper) in play that adds to the Draw…

Contrast this with Actions that give you extra draws on a one-time basis. If I play an Action or take an “optional action” that lets me draw cards…