How can I make squishy softbodies?

Imagine an extremely elastic balloon filled with air, if you press it against the wall with your hand its sides will expand relocating the air inside, or a water filled balloon. How can I simulate this? Blender softbodies is more like a deflated rubber ball filled with nothing that collapse inside itself on collision.

I’d love to know a better way, but the closest I’ve been able to get to that is using cloth physics on solid objects, and adjusting “pressure”, “tension”, and other sliders under the cloth sim. I’ve gotten some good results over the last few weeks, but I still haven’t been able to get exactly what I want which is very similar to what you described.
You can also play with “internal springs” in cloth sim, but most of the time I get closer to what I want without it.