Nothing much to report I'm afraid. :-(
I ran out of time to check the Winkles (Thomas, William, Elizabeth) I was checking Burslem parish instead of St John's Hanley for them so they could well still be in there.
I looked at records for:
St John's Burslem 1819 - 1828 and 1838 - 1841
St Peter's Stoke 1820 - 1825
St John's Hanley 1816 - 1827
Wolstanton 1815 - 1825
There were no Sawyer records at all in Stoke, Hanley and Wolstanton (or Winkles for the periods I looked in)
In St John's Burslem there were
30 April 1820 Hart. (Harriett?) D of Ann Winkle, Burslem, spinster
19 Sept 1820 Enoch son of Saml & Sarah Sawyer, Sneyd Green, f. collier
25 Oct 1820 Ellen d of Joseph & Mary Winkle, Cobridge, f. potter
9 June 1822 John son of John & Ellen Winkle, Cobridge, f. potter
30 June 1822 Harriet d of Ann Winckle, Burslem, Spinster
30 May 1824 George son of Hannah Sawyer, Sneyd Green, Spinster
6 Aug 1826 William son of Jho & Maria Winkle, Dale Hall, f. potter
9 March 1828 Aaron son of Thomas & Harriet Sawyer, Burslem, f. potter
and
19th April 1812 Ann d of Jesse & Elizabeth Winkle, High St Burslem.