Andy gave the official ruling.
You do NOT have to draw first, before you can play.
In the case of the Draw N rules:
The card states: “…..SHOULD immediately draw extra cards (IF NEEDED) …..”.
This goes into effect when the card is put in play.
Since it says SHOULD, it is a strong suggestion that it be done immediately (and at my table it is), but it is not an imperative (Not a YOU MUST).
So, based on Andy’s ruling, as long as you draw and play all the required cards, before the end of your turn, you can do so in what ever order you like.
The rule also says “IF NEEDED”, this is to rule out drawing more cards then necessary, I.E. playing a draw 5, drawing the 5 cards and then playing a draw 3. You only get the original 5 cards, not 8 cards.
Hopefully this has put to rest any question on when to draw cards.
And remember, if you feel strongly that you MUST draw first, just make it a house rule.