If making cushions, try using a thin batting instead of stabilizer. Yes I would be concerned about irritating baby's skin with the stabilizer, unless they have a singlet on - which most young mothers don't use these days. I also have issues with the embroidery thread on my skin, even when using iron-on interfacing on the back as the glue in that will irritate.
Welcome to the Board, you will find lots of help here if you ask.