I assume this is for an airplane since you said carry on and checking. My guess is that you will have to fold it somehow in any case unless you are flying First and FA hangs it in the closet.
I know this is not the answer to your question, but I am still going to give it.
I travel with a rollaboard that has a folding garment bag as a part of it (has 3 sections and makes 2 folds). I usually don't have creases from where the shirts were folded but they look not so good after coming out (though not bad). I usually give them a good shower press (turn on hot water in the shower, hang the shirts on the rod, close the door) - remove in 10-15 mins, and let dry. They will look perfect in the morning. Done many times. I hope nobody here is green but all my colleagues do the same thing - it is terrible for environment but everyone does it.
I never put put my jacket in the bag - instead i neatly fold it on top of my bag in the overhead bin or now that UA has unlimited upgrades, it usually gets hung by the FA in the closet.
Few more thoughts:
I prefer the rollaboard as it allows not to have to carry a bag (the laptop bag rides on top of it or gets hooked on the back) - helps with long runs through ORD and also makes the jacket less wrinkly (destroyed) from carrying a bag.
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