LDS lists a William Applegate born About 1800 Egg Harbor, Atlantic, married in New Jersey about 1828 to Hannah White.
1830 census William age 20-30 lived in Colerain Twp, Hamilton County, Ohio with one son. Next door is Richard Applegate age 40-50 with many children.
1840 census William lived in Colerain Twp, Hamilton County, Ohio with 4 sons and 1 daughter.
1850 census William age 49, born NJ, and Hannah age 46, born NJ, lived in Colerain Twp, Hamilton County, Ohio with their sons including Paul age 12.
1860 census A William age 58, born NJ, Sarah A age 49 born NJ, Mary age 22, Sarah A age 17, Newton age 13, Misandra age 10, Louis K age 7, and unnamed age 4/12 lived in Brookfield Twp, Trumbull County, Ohio.