There are several other Lawrence baptisms in Rushden -
Elizabeth 1783, Thomas 1785, Elizabeth 1786, Esther 1786, John 1789.
Thomas died as a baby in 1785. There's a John died 1824 Higham Ferrers aged 31, who left a Will.
The 1783 Elizabeth was only entered in the PR as "(blank) d (blank) & (blank) Lawrence", with the names only in the BT. Could that be Sarah?
William & Esther were married in Rushden in 1777, he was of Newton Blossomville, Bucks, while she was of Rushden.
The Buckinghamshire Baptism Index on FindMyPast has a daughter Jane born 8 Jul 1778, bap 21 Dec 1778!