Hello Brianthetaff
I would suggest you subscribe to a months trial of UK Genaealogy on line
http://ukgo7.com/register2016.html (£5.50) and use their huge database of Carmarthenshire records in your research.
I had a quick look there and found a marriage of a Walter Brown to a Jane King, in Pembrey, on 01/01/1771. Is this a likley match? It is the only one in the index of Carmarthensire marriages for this pairing from 1754-1837.
With such common names, there are lots of match for marriage of a John Griffith to an Elizabeth, pre 1837. If they came from Pembrey (or at least Elizabeth did), there are only 2 matches. One for a Elizabeth Richard on on 24Nov1824 and one for Elizabeth Gower 25Mar1828. Are these too late? What were the relative ages of Elizabeth (1814-1886) and her husband (name ?).
Have you found them in the 1841 census. From census records what year was their oldest child born. This should give you a rough idea of what year they married.
Find my past also has parish records, including those from Carmarthenshire
http://search.findmypast.co.uk/search-united-kingdom-records-in-birth-marriage-death-and-parish-records you could try searching
You could also try searching Freereg
http://freereg1.freereg.org.uk/cgi/Search.pl , though their records are incomplete.
If you post details (e.g. names and ages of spouse and children and where they were living) from your census finds and any other relevant details, others may be able to help