You have 2 cards in your hand, hand limit of 2 is in play as well as draw 5 and play 3. On your turn do you draw 5 (giving you 7 in your hand) and then choose which 3 to play, then discard the remaining 2 to end your turn with 2 in your hand? Or do you draw one and decide between the 3 your holding which to discard, then repeat for the next 4 cards drawn?
My wife and I had trouble agreeing how to interpret the order
You complete each action in turn, so the first interpretation is correct. It's "Draw 5" so that is what you do. This is important because subsequent plays can change the rule. The Hand Limit specifies at the END of your turn, so it can easily be gone before you are finished. You can, with the right card draws, play through a lot of turn and just before the end trash the hand limit or replace it with a higher one.