No, what happens before the discard is that the “draw a card” ability gets added to the end of a queue of things to do after Shadow Blade is over.
Here, let’s play Shadow Blade on an enemy Technician Older Brother.
at Time 1:
Currently running: the part of my main phase where I choose which action to take, like hiring a worker or playing a spell.
Currently in queue: nothing
My gold: 3
My units and heroes: Vandy Anadrose
My hand: Shadow Blade, Dark Pact
My opponent’s hand: 5 cards
My opponent’s Older Brother: 2/2 with 0 damage in the Technician slot.
I pay 3 gold and put Shadow Blade on the table.
at Time 2:
Currently running: Shadow Blade.
Currently in queue: nothing
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 5 cards
My opponent’s Older Brother: 2/2 with 0 damage in the Technician slot.
Well, to actually run Shadow Blade I guess we should do the first part of Shadow Blade. I’ll target the Older Brother and deal 3 damage to it.
at Time 3:
Currently running: Shadow Blade.
Currently in queue: nothing
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 5 cards
My opponent’s Older Brother: 2/2 with 3 damage in the Technician slot.
At this point, after the first part of Shadow Blade, the game notices that the Older Brother should be dead! So it dies. Immediately when it dies, we enqueue the Technician ability.
at Time 4:
Currently running: Shadow Blade.
Currently in queue: "My opponent draws 1 card"
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 5 cards
My opponent’s Older Brother: in my opponent’s discard pile.
The game is now satisfied that the state does not need fixing (through e.g. the killing of things that ought to be dead), so it lets us proceed to the next part of Shadow Blade. Shadow Blade did kill Older Brother, so my opponent discards a card.
at Time 5:
Currently running: Shadow Blade.
Currently in queue: "My opponent draws 1 card"
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 4 cards
My opponent’s Older Brother: in my opponent’s discard pile.
We ran out of things to do for Shadow Blade, so we’re done casting the spell. It goes to our discard pile, and we move on to the next thing in the queue.
at Time 6:
Currently running: "My opponent draws 1 card"
Currently in queue: nothing
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 4 cards
My opponent’s Older Brother: in my opponent’s discard pile.
So my opponent draws a card!
at Time 7:
Currently running: nothing
Currently in queue: nothing
My gold: 0
My units and heroes: Vandy Anadrose
My hand: Dark Pact
My opponent’s hand: 5 cards
My opponent’s Older Brother: in my opponent’s discard pile.
We ran out of stuff to do in the queue, and we’re not doing anything right now, so the game moves us along. In this case it takes us back to “the part of my main phase where I choose which action to take, like hiring a worker or playing a spell.”