Teresa, I can see that the Haines family is quite a challenge because there are so many of them. I have some Ohio research resources and War of 1812 resources, so I did a little digging. Have you sent for William's Service Record from NARA? There are four William Haines/Hains/Hanes/Haynes in the War of 1812 Index, but the one that is probably him is the William Haines who served in Captain Joseph Evan's Company of Ohio Militia, as a private. This company served from Feb 24, 1813 to March 24, 1813. The company was made up of mostly Champaign County men, which at that time also included Logan and Hardin County. They guarded the government stores of supplies at Fort Findlay. Also in that company were Samuel Haines, Thomas Haines, and Abraham Painter. I found more info on this Samuel Haines, and can say that he was definitely not a brother of Thomas or William, although he could have been a cousin. Many of the veterans of the War of 1812 who were still alive in 1850 and 1855 applied for Bounty Land Warrants. These were available at two times - 1850 and 1855. Those applications are also available from NARA, and they are a gold mine of information, if your ancestor applied.
I checked my resources for early Ohio marriages, and I found nothing for William Haines and Nancy Hatfield. This makes me suspect that they could have been married in another state.
I was wondering what Hardin County records you might have already looked at. Have you looked at early tax records? Deed records? Will records? Death records? Cemetery records?
Sarah Haines is my husband's third great grandmother. She was born 8 January 1820 in either Ohio (per the census records, or in Culpepper County, Virginia (per a family history from my husband's Great Aunt). However, per the 1880 census, her parents were both born in New Jersey. She died in Carroll County, Indiana 25 March 1899. She first married Benjamin Palmer in Logan County, Ohio in 1843.
Her oldest two sons were born in Logan County, Ohio, then she and Benjamin Palmer moved to Conway County, Arkansas, where sons George and Alexander were born. Her husband died very young, about 1852 from typhoid fever. Her second marriage was to John Day, a widower, in February of 1853. In the 1850 census, John Day is living in Hale Township, Hardin County. Hale township is adjacent to Taylor Creek township. I think that I found your William's family in Taylor Creek township - is that the right family? Sarah and John were married in Hardin County. I am only guessing, but it would seem that a widow with four young children would move close to relatives after the death of her husband.
I do not know Sarah's parents or any siblings. However, three of Benjamin Palmer's siblings married Huffman's who were brothers and sister. The Huffmans were the children of Kindle and Rebecca Huffman. The reason why I am telling you about the Huffman's is that they have a New Jersey connection.
Kindle Huffman is listed on the 1830 census for Evesham, Burlington County, New Jersey, directly below Thomas Haines. Abraham Haines is on the same page also. So my theory is that the Huffman's and the Haines knew each other in New Jersey, and possibly travelled together or separately to Logan County. Evesham is apparently where the early Haines family lived before branching out to other states.
Can you tell me anything about an approximate year of marriage for William and Nancy, even though you haven't found the marriage record? Do you think that this was a second marriage for him? I see in the 1850 census, that there is a Mary in the household, but no Nancy.
Do you know anything about their religion? I understand that the Haines family in New Jersey were members of the Quaker church. I do not know if my Haines family left the church or remained Quakers when they moved to Ohio. I would like to find out more about researching Quaker records.