I have a Thomas L. Taunton (3 Oct. 1881 - 28 July 1935) married to a Martha (not Mary Elizabeth) Spencer (11 July 1884 - 31 Dec. 1967). They were from Union Parish, LA. Would this be the couple you're looking for?
I show Thomas L. Taunton as the son of Jesse Stephen Taunton (1849-1919) and Martha Elizabeth Wood (1858-1919). Jesse was the son of (John?) Tillman Taunton (1820-1889) and Mary "Polly" Brown (1822-1884). Tillman was born in Georgia and somehow connects to the rest of the early Tauntons who arrived in Georgia around 1800 from the Edgecombe/Pitt Co., NC area.
There is no proof that the southern Tauntons were ever in Massachusetts, although some people assume this because of the town in Massachusetts named Taunton. Taunton, MA, however, was named after the town of Taunton, England, NOT after a family named Taunton. The Tauntons in TX, LA, AL, MS, GA, and FL all trace back to the Edgecombe/Pitt Co., NC area around 1735. It's possible that this is when the family first arrived from England, or perhaps they migrated down from Virginia; there was a Robert Taunton who arrived in Virginia in the 1600s, but nothing more is known about him.
Hope this helps! Please email me directly if you'd like any more info.