I'm increasingly worried that the shroud system doesn't tell the story you want it to. If the assassin is the guy who formulates and executes precise plans, a trickle of "free" damage resource at the start of each turn seems contrary to that -- it looks more like a battlemaster or monk with a few extra bookkeeping hoops to jump through. In order for the player to feel like they're consciously plotting, plotting needs to be a conscious decision with some sort of meaningful opportunity cost (but, of course, balanced to be worth it). A couple of brainstorms:
One Step Ahead
Even as you finish one move you are busy plotting the next one. When you roll initiative, and at the end of each of your turns, you can choose a tactic for your next turn from the following options.
Execute. Choose a creature you can see. The first time on your next turn you hit that creature with a weapon attack, you deal [a lot of damage], or [even more damage] if it is vulnerable. If its hit points afterwards are less than or equal to [that damage], you can immediately reduce it to 0 hit points.
Cripple. Choose a creature you can see. The first time on your next turn you hit that creature with a weapon attack, you apply [some condition that triggers vulnerable].
Riposte. Choose a creature you can see. The time it makes a melee attack against you before the start of your next turn, it has disadvantage on that attack, and if the attack misses, you can make a melee weapon attack against it as a reaction.
Reposition. On your next turn, your speed is doubled, your movement doesn't provoke opportunity attacks, and you have advantage on Dexterity checks.
Hit List
Whenever you roll initiative, you create a hit list, an order in which you intend to kill your enemies. You can put on this list any number of surprised creatures you can see, plus a number of non-surprised creatures you can see equal to your Intelligence bonus (minimum of one), in the order of your choice. Once per turn, when you hit the first creature on your hit list with a weapon attack, you deal [some damage]. If the creature's hit points afterwards are less than or equal to that damage, you can immediately reduce it to 0 hit points.
When the top creature on your hit list is reduced to 0 hit points, you can cross it off your list, and the next creature on the list moves to the top.
As an action, you can discard your current hit list and make a new one.
And a parallel thought for spicing up Expertise, making the assassin a sort of "skill wizard" to the rogue's "skill sorcerer":
Expert Planning
Whenever you finish a short or long rest, you plan which of your skills and tools will be most necessary for the coming action. You can make a number of plans equal to [some level-based formula]. For each plan, choose a skill or tool in which you are proficient. You can choose the same skill or tool for multiple plans. When you make an ability check using a skill or tool for which you have a plan, you can expend the plan to reroll the check, doubling your proficiency bonus for the reroll, and use the second result if it's higher. Alternatively, if you could take an action to make an ability check using that skill or tool, such as the Hide action for Stealth or the Search action for Perception, you can expend the plan to take that action as a bonus action, doubling your proficiency bonus for the check.
Plans last until you expend them or you finish your next short or long rest.