Are they too small? Try another brand. Cotton should be soft. I have some 100% Italian cotton socks and they are the softest ones I own.

Freeze them after washing them. When I was a kid we did not have a dryer, and dried clothes on a clothesline. One thing I loved about the winter was how soft my clothes felt after they froze on the line before drying.

Try some different brands, there are definitely some higher quality cottons socks compared to brands like Haines.

…. we’re confident these socks aren’t being stiffened with bodily fluid?

Are you sure they're cotton? Synthetics tend to be more stiff than natural fibers like cotton.

I haven't had stiff cotton anything since about 1978. Companies have been pre-washing cotton since at least the mid-70's to make it softer.

Try washing and tumble dry on low, without completely drying them.