Why does the fabric seems to have an odd, almost chemically smell?

That unusual smell (that only some people notice to an unpleasant degree) isn’t a chemical on the fabric at all as we do use 100% USDA Certified Organic Cotton. It is simply the natural “earthy” plant odor from unprocessed cotton that hasn’t been treated, bleached, dyed or deodorized.  It is completely non-toxic and will dissipate naturally over time.

It may smell more intense when wet (post-washing machine) the same way many other things smell stronger when wet, such as dogs or grass.  Most of this is due to the oil from the cotton seed which is present during ginning; and with all oil-based substances, can linger longer on fabric.  One way to help dissipate it faster is by soaking the pillow or case with a little white vinegar and baking soda before washing.

Do know there are definitely some people who notice this plant smell more and interpret it as offensive than the average person. It’s a small minority but probably an indicator of super-smellers, like those persons who are super-tasters. 
Also, sometimes a stronger smell may be present on some items and not on others but this is due to those bundles of fabric that were processed longer ago from the field and/or knitting facility vs those more recently made.