So far I haven't seen this response here, (and I could be waaaaaay off) but it almost sounds like you have a "very" hot iron and a fabric blend of some polyester or acetate, perhaps mixed with cotton???
I typed "very" as it's waaaaay too hot for a blend but not for 100% cotton. You might need to do a fabric "burn test" to determine what it is. Carefully... try...
http://www.fabricmartfabrics.com/Burn-Test-Chart.html
If I'm not sure, I usually do the test to determine animal (wool); vegetable (cotton) or man-made (polyester, nylon, rayon, etc.) although ironing, I can usually tell from the "smell".
Cotton wrinkles should come out with a hot steam iron. Perhaps a little fabric finish will give the cotton it's "hand" back.
But to be sure,
I WASH EVERYTHING; iron; and spray with fabric finish (not starch).
Good luck.