In 1841 census a William Cawood aged ca 25 and wife Sarah ca 20 (but ages rounded down in 1841) both born Yorkshire were living in Headingley, Leeds with children Elizabeth and Edward. In 1851 they were there, aged 36 and 34 with additional children Mary, Sarah, William and Thomas. William born Otley, Sarah in Yeadon and children all in Kirkstall.
Baptism at Otley 23 Feb 1817 of William s/o Elizabeth Cawood would seem to be the one. But there were several Elizabeth Cawoods around in Otley at the time.