My heart goes out to you. There is nothing more frustrating, more aggravating than having new bees swarm. I've been there.
Bees swarm for several reasons, and in my opinion, they swarm due to congestion, not necessarily crowding.
I define congestion as competition between the space needed to store in-coming nectar and the space required for the queen to lay eggs. When there is too little cell space for both eggs and in-coming nectar, the bees sense the need to make swarm preparations.
With new foundation, there is little drawn comb, and thus, even adding a honey super won't help. You need drawn comb. You need cell space, not just extra room. As soon as they start drawing out the comb, the competition for cell space is reduced. But you got to get them drawing out the foundation.
However, the bees make swarm preparations before they actually swarm. Adding a honey super may be too little too late.
One trick I was taught was to weave frames of foundation between frames of drawn comb, especially emerging brood. This seems to make the bees draw out the comb faster than hoping they'll feel the need to expand horizontally.
Hang in there.
Grant