They do make briefs with the material looped over the elastic and then sewn down so you can't feel the elastic against your skin. However they cost a bit more.
Also material/elastic can feel coarse against your skin if it is very dry. Try using some moisturizing cream on the area.
I have no idea what you're talking about that supposedly make you itchy. I don't have any problems with underwear unless it's boyleg, they give me wedgies and feels like i'm wearing g-strring.