You don’t really get to choose the order that units die in. They all die simultaneously, you just get to choose the order they go in your discard pile. Any effects which are triggered by units dying will happen after everything has died. So all units go to the discard pile, then you draw a card.
If there are multiple effects triggered from units dying, you can choose what order to resolve those effects in.
ETA: Yes, technician draws happens immediately, and so you can discard the card they drew with Carrion Curse. It doesn’t really break the asynchronous nature of the game, as there are no choices to be made by the other person. If the game was implemented online, it can draw a card for your opponent and then show you their hand without needing that person to respond.