Hello Carol. I only have the inscriptions for those buried at Witton le Wear so your Thomas Smith and his wife would have been buried at Hartlepool or is last place of residence.
There is one Witton le Wear gravestone for Smith but it is in poor condition. Most of it is very weathered, but here is what can be read.
In memory of Christopher Stephenson, late of the C?


?, Durham, who died August?


?5 years.
Likewise to preserve the memory of Hannah Smith, wife of Robert Smith of Witton le Wear and daughter of


???Dixon of To (possibly Tow Law)
who died January ? 1857 age ??
Also of Judith Ann, daughter of the above named Robert and Hannah Smith, who departed this life in infancy.
Adam