Hello May
The image of the 1851 census look like Leamington to me.
If you Google Leamington, Dorset, you get several family trees from the 1600's and 1700's, with people born Leamington, Dorset and two roads in Dorset - one in Weymouth and one in Bournemouth.
It seems therefore it was a place in south Dorset once, probably "swallowed" by a larger town, although which - don't yet know!
Hope this helps