I hope this helps to clarify a little (though lots of holes in the records I have found, and you possibly already found this). I would very much like to sort out these families!
Thomas Jesson - a gold and silver smith and scale maker - Thomas Fox Jesson baptised 1779 Dublin son of James Jesson (gold and silver laceman of Dames Street) and Rachel Smith. Used both Thomas Jesson and Thomas Fox Jesson. Named after Thomas Fox (d ca 1784) gold and silver laceman of Parliament Street, former employer of James. This Thomas married Letitia Robinson in 1818. I have found 5 children including James. I believe this is the one that married Eliza Darling in the link you posted. The last record I have found for him is 1837.
Thomas Jesson - a pawnbroker - Thomas Richard Jesson born about 1780 son of Robert Jesson (probably the pawnbroker listed in 1790 newspaper ads). This Thomas married Maria Jane Clavel in 1809.
I don't know if the families are connected. The only link is the common use of the names James, Thomas and Richard. I cannot find parents for James or Robert. I can trace Thomas Fox Jesson and his family down but not further back than his parents. He had brothers Richard (joined the British Army), Horatio (also a silver smith) and James (an auctioneer).
Sue