If you are running out of space and think an airlayer may take too much time, I recommend trying to just root cuttings. I have done this with my standard papaya's and Mountain papaya (pictured below) and always find it relatively easy.
I can't find a good picture of the fully developed roots, but here is my method (started April '23) and one of the spare plants that I sold (November '23). I kept another one of these plants for myself that was ~2x the size as the one pictured, but I recently gave it to a friend and never took a photo.
I used a seed starting potting mix with a touch of perlite. I've used this method on more plants with some success, though I would recommend a much thinner layer of perlite at the bottom. The one pictured was my first attempt with this method.
I cut drainage holes in the bottom and along the side of the bottom of the cup. Nowadays I will also wrap longer cuttings with grafting tape or put another cup on top of small cuttings (with ventilation) to create a humidity dome.
With this, you can focus on shaping your remaining potted papaya plants and not have to worry about the impact of a developing airlayer.