The 501(c)(3) status will not transfer. You will be starting a new organization with a completely different entity. So you will need to file and pay the fee to apply.
The same thing happens if you are a PTO that incorporates (after getting the 501). The incorporated PTO is a new legal entity, so the 501(c)(3) doesn't transfer.
We are currently a PTA looking to change to a PTO. We now have 501-c3 status and I was wondering if we will need to reapply for that under our new name and by-laws or is it strictly a name change? Does anyone know how to go about this?