Hi Carole,
All Thomas ADAMSON and Ann CHAMBERS children have birthdates and baptisms as shown on IGI.
George ADAMSON 1st son of Thomas ADAMSON, labourer native of Redmarshall, by his second wife Ann CHAMBERS, native of Houghton le Spring
Mary, 1st dau, details as above
Eleanor, 2nd dau, details as above
Thomas, 2nd son, details as above
John, 3rd son, details as above
Jane dau. of Thomas and Ann ADAMSON late CHAMBERS, Farmer and Innkeeper, Billingham
The marriage between Thomas and Ann 19/11/1802 gives no indication as to widow or bachelor. Both left their mark and witnesses were William Holborn, Mary Atkins and Mary Marshall.
Witnesses to the marriage of Thomas and Elizabeth FAIRBARN 14/5/1794 were James Adamson and Mark William Warton, Parish clerk
I'm afraid I couldn't find baptisms for Ann or Elizabeth Adamson
This could be relevant though - Ann natural daughter of Ann CHAMBERS was born and privately baptized 16th May 1796, Billingham
There are two children of Robert ADAMSON, fellmonger, and Grace (nee HEADLAM) - John born 10/10/1797 and privately bapt. 24/10/1797 and Mary bapt 6/10/1799
There is Elizabeth ADAMSON born 21/2/1811, bapt. 26/5/1811 illegitimate dau. of Ann ADAMSON of Redmarshall - I guess this is too late for the Elizabeth you wanted.
Witnesses to the marriage of Robert KNAGGS and Ann ADAMSON 17/5/1823 were John Holburn, George Adamson and Elizabeth Adamson
Witnesses to the marriage of William MADDERSON and Elizabeth ADAMSON 24/5/1825 were George Adamson and Thomas Madderson
Finally Elizabeth ADAMSON of Billingham, late FAIRBARN wife of Thomas ADAMSON, labourer, died 7/9/1801, buried 9/9/1801, aged 31 (or 37) Billingham
I'm sorry I couldn't find the two you were most interested in but hope this helps a bit
Nannawilk