Q: If I play an Action or use a “Free Action” that lets me draw cards, and then I increase the Draw rule, do I still get to draw extra for the Draw rule increase?

A: Yes. The draws caused by cards that have one-time effects (actual Action cards, or “Free Actions” granted by New Rules in play) are not to be considered part of your Draw (they are “extra”) so they don’t count towards how much you have Drawn.

Common examples include Jackpot, Mystery Play, Draw 3 Play 2, Recycling, but there are so many, we have given up on tagging and listing them all.

So, for example, if the rules are currently Draw 2, Play 2, and a player decides to play the Action Jackpot, which lets them draw three extra cards, then they increase the Draw rule to Draw 4, they still get to draw 2 extra cards. Though it may appear that they have drawn a total of 5 cards during their turn, the 3 cards they drew because of Jackpot do not count against the required Draw for their turn. They’ve only officially drawn 2, and now need to draw 2 more to bring their total to 4, as per the Draw rule increase.

Likewise, if someone utilizes the Free Action Recycling, allowing them to discard a Keeper in play to draw extra cards, those extra cards are not counted as part of the quantity of cards they have “drawn on their turn” as required by the official Draw rule.

Contrast this with Rules discussed in the question linked below which are formulaic modifiers to the actual Draw (or Play) count itself.
See also If I play a New Rule that’s not a Draw or Play, but increases the Draw or Play…