Eilleen
If Annie's Father was William Moore, Labourer and if he was a Farm Labourer, Annie could of been born Gosberton and he or family moved on, end of the Season.
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Non-Conformists
I wonder if any Nonconformist (NC) records survive at Archives for Gosberton 1850 to 1860 to check for any Moore births?
Baptists had a Chapel at Gosberton from 1666. There were also two Anabaptist families who met twice a year at one point, probably in a house.
By the time of Annie Moore's Birth, Wesleyan Methodists and seen references to Primitive Methodists one book mentioned New Methodists. (Primitive Methodist in some places was usually after Wesleyan Methodists had a chapel).
I wonder if there were any Catholics living at Gosberton? In some places c.1850 Catholic Registers are still at the nearest Catholic Church, or their Area Office. But some Archives have filmed them.
Quakers seemed to be at Spalding.
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Some Lincolnshire Male & Female Moore Baptisms found, Father William Moore (limited by mileage radius):-
1859 Annie Florence D of Wm & Mary Moore, Burton Coggles, Farmer.
1857 Emily D of Wm & Mary Moore, as above.
1857 Marion D of Wm & Mary Moore, as above.
Burton Coggles was about 20 miles from Donington.
1858 Wm Christopher S of Wm & Ann Moore, Dunston, Cordwainer.
1860 Betsey D of Wm & Ann Moore, Dunston, Cordwainer.
1857 William S of William & Hannah Moore, Stamford.
1857 Sarah Ann D of Wm & Eleanor Moore, Castle Street, Labourer. Stamford Register.
1859 Frances D of Wm & Eleanor Moore, Brooker Court, Labourer. Stamford Register.
1860 Harriet D of Wm & Frances Moore, Bessingham, Labourer.
1855 William S of Wm & Hannah Moore, Syston, Lincs.
1860 Sarah Ann D of Wm & Hannah Moore, Syston, Lincs.
Other Brief / Summary Notes
1839 Wm Moore, Lab'r & Mary Ann Smith, Morton.
1855 Wm Francis Moore & Charlotte Walker, Labourer, Gedney.
1858 Wm Moore & Ann Curtis, Farmer, Coningsby.
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After your Annie Moore married T C Exton in 1875, it would be interesting if an Exton was present as a witness at any Moore, Lincolnshire Marriage?
Mark