From your description, it sounds like you may actually have
Sphagnum Moss instead of Sphagnum Peat Moss. Canadian Sphagnum Peat Moss should be completely decomposed and you shouldn't be able to see any "fronds" or even tell that is used to be a plant. If you indeed have Sphagnum Moss, I would not use this under your sand, been there done that a long time ago
Is it possible for you to include a pic of the "fresh peat moss" you plan on using?