Does Polyurethane foam also stabilize soil?
All buildings settle as they are constructed. What we attempt to do is minimize the settlement and have the building settle uniformly. 

Unfortunately you live where the soil moves (expands) due to moisture. This movement is caused by the wetting AND drying of the soil. As the soil gets wet, it expands...and vice versa. Using a “filler” in the soil, will not solve this problem. 

Polyurethane foam in soil works if you have “weak” soil conditions. It will not work in expansive soils. I live where the soil is “mush”.  When we want to stabilize the soil we’ll use something like poly foam. When we’re done the soil is stabilized (no movement).

You can inject poly foam into the soil, but the soil will still be expansive. As soon as it rains or the soil dries out, it will expand or contract and cause movement in your house. 

You should use drill piers or drive piling through the expansive soil. Then, when the soil starts moving, it won’t matter. 

For new construction, we’ll often over excavate the site and then install a gravel base 6-7’ deep and extending 4-5’ past all the footings. This is expensive, but gives a “uniform” base so the house settles uniformly and we can calculate the amount of settlement based on the size (height and loads) of the house. Obviously you don’t have this option and I don’t think it would work where you live anyway because the soil under the new base could expand and cause problems up through the new gravel fill.