Obviously the sprint for weapons and ignoring tasks at the beginning of every round is what we're trying to get away from and the only way to do that is to not have weapons available to find at the beginning. So here's my idea, and bare with me it takes a bit of explaining. Also all these little rules can be subject to change, this is just the premise.
WEAPONS
With the new armory room, you make a random selection of weapons available to pick up on the wall of the room and the door is locked. Sometimes all of them will spawn, sometimes only one, etc. To unlock the door you need a 4 digit code, much like the crates work right now, except each player only gets 3 attempts to unlock it. If one person uses their attempts, then others can still try. This will make it so no one can just waste attempts to lock the door permanently. Now at the beginning of the round no one can interact with the keypad until at least one task/subtask is completed, making a rush to the weapons pointless with tasks as a priority, and a risk/reward for the dissident (if they want the guns, they need to help complete a task or just wait). 3 attempts to unlock the door is obviously very difficult with no info so here's the second half of this idea.
EMPLOYEES
So for employees, you get one digit of the code for each task/subtask you complete. This would add to the social aspect because you may get 2 digits, and compare them to a fellow employees digits and be able to figure out the code, while a dissident may do the same or give you incorrect numbers so you can't get the weapons.
DISSIDENT
For the dissident, I'm thinking each kill gives you 2 digits of the code because usually there's more tasks/subtasks than people to kill.