I've had conditions as you describe with a lost clipped queen. (You have to walk around the apiary VERY carefully - I've been inches from standing on one!)
With a large colony, leaving 2 cells in the hive is planning to have a swarm.
What I would do is to go through the hive and select one or two good open queencells with the larvae floating in royal jelly. Mark on the frames where they are. I would remove the rest. A week later, the queencells would be sealed and there would almost certainly be more. I would consider using one of the 'original' sealed queencells by putting it in a nuc on it's frame with some shaken in bees and a frame of stores. Close up and put in a cool dark location for 3 days. (Some books say use a cellar, but as most of us don't have one of those, the back of the garage or behind the shed where the sun don't shine is fine). The remaining 'original' queencell would be retained in the hive. All the later - emergency - ones would be destroyed by jolting the bees off the frames first so none are missed. (Don't jolt the frame with the wanted queencell). We would then expect the queen would emerge a few days later and then mate over the next 2 - 3 weeks. If you leave one crummy emergency queencell, the wanted virgin queen will probably swarm leaving the younger queen to take over.
The nuc is your insurance policy.
If you want to have two strongish colonies, you could consider an artificial swarm a week after the attempted swarm, with one queencell on it's frame on the old site which the flying bees will return to and the other queencell remaining in the hive on a new site a few feet away.
May your bees read the same books as you do.