You can certainly achieve a good part of your aim by using Puffy Foam but you do, I understand, need to consider altering the thread stitch. I believ that Embroidery Library has a technical page on the subject.
Probably could try using some batting, but will possibly only work if you are doing a Redwork design.
Hi Nancy, the only thing I can think of is to build up the designs, by adding different layers. There are some 3D designs on Emb Lib that show you what I mean. Otherwise you could try doing a few of the same design and build them up like they do with papercrafting in Scrapbooking etc. Love and blessings Chris