Thomas Williams - Thomas Williams, born 1791 in Grafton, New Hampshire

I have been working on the tree of Richard Williams, "Father of Taunton" for a decade. I have stuck to very reliable sources, especially those discussing Mayflower descendants.

There is a branch beginning with Rufus Williams that I did not add. I have been verifying that there are sources for this branch. Here is the descent from Richard Williams:

1. Richard Williams, of Taunton
2. Benjamin
3. Benjamin
4. Rufus 1723
5. Samuel of Grafton 1746

There is an excellent online genealogy for Samuel Williams, born 1746, of Grafton. I feel confident that the above descent to Samuel is correct. At this point however, we need to find multiple reliable sources for a Thomas Williams, born in Grafton in 1791 being the son of Samuel Williams, born 1746, of Grafton.

Further, the Geni tree has Samuel Williams, born 1817 or 1818 in TAUNTON, Massachusetts as the son of Thomas Williams, of Grafton. This is questionable since the Williams of Grafton remained in Grafton for generations.

Further, whoever Samuel Williams born 1817 in Taunton is, what is the evidence that he is the Samuel Williams who shows up later in New Zealand and eventually in Australia. If the Australian Williams have male line descendants, their DNA could be tested. My Williams cousin has done a Y-DNA test and I believe there are others, likely in the FTDNA Williams surname group, who are proven descendants of Richard Williams, of Taunton.

So I am asking for help in finding actual evidence for generation #6 and #7 below and separately for a Samuel Williams of Taunton ending up in Australia.

6. Thomas 1791 Grafton
7. Samuel 1817/18 TAUNTON