I sew little dresses for charity and have found that pressing with Magic Sizing will flatten most center fold creases and any other creases. Or if you do wash the fabric, use Magic Sizing to iron it and it will have the feel of new fabric.

These sprinkle caps are still sold through The Vermont Country Store, although they are plastic now.

I spritz (or sprinkle) the unwashed fabric, let it sit for a few moments to absorb the water, then iron. For large pieces of fabric, I, too, spritz, refrigerate overnight then iron. For really stubborn wrinkles, I use Magic Sizing spray. I am a confirmed non-washer (unless it is a red, black or navy fabric).

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.