Any grass seed needs to have soil contact to grow. It won't do much on dead sod.
Peat moss under the sod is not the way to prep the soil. Leaf compost or screened topsoil mixed with the existing soil so the sod makes good contact is what's needed. Maybe a little starter fertilizer mixed in.
Agreed, peat moss wasn't the way to go. In every sod project I've undertaken, I've put down several inches of compost over the entire area, worked it in with a rototiller, rake it out, roll it, wet it, then lay the sod, roll again, and soak generously. I've never had a bad experience with this process.
The ground needs to be wet when you lay the sod or you dry out the roots even more, which stunts or kills the sod.
As for seeding over the sod, you might be able to get the seed to germinate, but you're still going to have bare spots for a good long time until the seed grows and the grass multiplies, many, many months. If you simply cut out the patches and lay in new pieces of sod, you will have a completely green lawn from the get-go. It isn't hard to cut little patches in, just be gentle as you work, and as was stated earlier, get yourself a crappy, but sturdy knife from a thrift store.