High alkaline detergents cause natural fibers like cotton swell, and under certian water conditions hard water minerals become can become trapped in those fibers. This results in the harsh, rough feeling many complain about when using detergents high with sodium carbonate. Arm & Hammer comes to mind as well as some BOL detergents.
Laundry washed with soaps had a soft and fluffy feeling, due to the fats and oils from soap. As detergents began to displace soaps for laundry, many housewives missed that soft feeling, hence fabric softeners came along. This new invention was pretty much nothing more than fats/tallow/waxes suspended in water. Fats/tallow are also what is in soap... (see where I'm going with this?).. Today's newer TOL fabric softeners are based more on cationic surfactants, but there are plenty of fat/tallow based products out there. Sure test is to allow the product to sit in a cold area for a few days. If it separates into soilds and water, it is pretty much tallow based.
As for cotton having "oils" the only oil one is aware of from cotton is cottonseed oil, but that product is made from cotton plant seeds.
L.