Q: Do I have to observe the current Draw rule entirely before I can play any cards?

A: Yes, on all turns you are supposed to observe the Draw rule before the Play rule, though it sometimes happens, if you change the Draw rule during your turn, that you may have to draw additional cards in the middle of your turn.

(Unless you have a Meta Rule that says otherwise.)

Note: Many of the Meta-Rules discussed in the linked article don’t actually exist. In particular, the one which would apply here, Quota Style Turn Order does not exist. Neither does Early Limit Compliance.