In the article it says he was in the Llanelly Workhouse, and an article of a report from the workhouse says he was absconding.
All census records for them, give very different ages and birth years. From the most reliable in 1881, he was 2 years younger than his wife, who I know was born in 1834. Therefore, he was born in 1836, in Newcastle Emlyn, Cardiganshire, so that side of the river. The marriage certificate in 1855, says his father was John Williams a Labourer.
It may not be the right Thomas Williams, but he is the closest I can find. Born 1836, Cardiganshire side of the river, but still regarded as Newcastle Emlyn, with father John Williams and a Labourer. I am asking, with a hope of finding more information to prove, or disprove my theory. It is a shame the 1861 census is missing, as that could have provided more information.
This is the family without Thomas:
1871 Census: 7 Gosport Street . Laugharne
Margaret Williams . Head . Married . 34 . Mariners Wife . Llansadwrnen
Esther Williams . Daughter . Single . 14 . Laugharne
Elizabeth Williams . Daughter . 6 . Scholar . Laugharne
Margaret Williams . Daughter . 4 . Scholar . Laugharne
Thomas Williams . Son . 2 . Laugharne
GBC/1871/5495/0018