The Rossiters are indeed hard to research.Unfortunately,I only have access to online records as I live in the UK.I'm not familiar with the stories but it is credible as a lot of people from England and Ireland emigrated to the US.
I have John and Nancy being born in Pennsylvania and ending up in Stewart County, Tennessee.It's possible that John & Catherine were either born in Pennsylvania or possibly in Ireland.Unfortunately,I don't have enough information to make a judgement either way.
I have a number of Rossiter families in Pennsylvania prior to 1800.Do you know please what part of Pennsylvania John & Nancy came from? Did Nancy's John have any siblings?
Hopefully,by getting small pieces of information I might be able to identify the family in my database and link them all up.