So it's not my turn but I have used an ability to draw up to 16 cards (max hand size I've been told). When it becomes my turn am I just not allowed to draw? Or when I draw do I have to discard down to 16 immediately?
Your hand is never allowed to go above 16 cards. If you reach your draw phase and your hand is at 16, you draw zero. If it's at 15, you draw 1, etc.
Only at the end of your turn, during the discard phase, do you have to have your hand down to 8 cards.