You may already be aware of them, but there are records for what may be your William and Elizabeth in the Famly Search database [https://www.familysearch.org]. ; If you create a (free) account there, you can find William under ID number LLC6-JPK, and can access the records for others easily through his record, as you will see.
The record shows only two children, the Sarah whom you have as being born in 1850 (though it says that she was born in Scotland) and an older child, William, who is shown as being born in "Ardoe, Tyone" on 16 March 1842 That is a typo, though (or two typos), because there is a document attached to the record which says that it was Arboe, in CountyTyrone (which is east of Cookstown, on Lough Neagh). I think the records may have been created by descendants of William (the son), because they then follow him to America, where he later married.
In each record at FS, you can see the screen names of those who have created or modified the record, and can send them private messages, so you might be able to get more info by contacting them (and perhaps also locate some relatives that way).
The surnames in this case seem to indicate that the family was not Catholic, but it's not always easy to be sure, and Catholics were numerous in that area. If they were Catholic, then the parish records for the Catholic parish of Arboe (in irish, Ard Bó, meaning roughly "cow heights") should be useful, and they are available online, going back to 1827, at this site:
https://registers.nli.ie/parishes/0181
Arboe is also the name of the local civil parish (an administrative district), about which you can find more info here, including further links to info about the various townlands within the civil parish:
https://www.townlands.ie/tyrone/arboe/
On the page for each townland, there are also links to records for that townland from the 1901 and 1911 censuses, as well as Griffith’s Valuation (mid-1800's), if they’re available. It's clear from the maps you will see at those sites that the Catholic parish is larger than the civil parish (a not uncommon occurrence), and part of the Catholic parish is actually across the border in County Derry.
If you're able to make use of the Catholic parish records, then each baptismal entry will usually indicate the townland in which the family lived, so you can combine the two sources and see where they lived (and possibly access the other links mentioned above).
If the family were Protestant, then the Church of Ireland or Presbyterian church(es) in the area should have records as well, and they often go further back than Catholic records do. It's likely that the C of I parish will also be called Arboe. Protestant records also frequently refer to the townland in which the family lived.
