Hi,
I know you have said that Elizabeth was born in Warwickshire, but do you know which parish? You might be able to get this information from another census perhaps.
You could then post your message on the Warwickshire page and somebody could look her up in the parish register.
You also don't say where the children were born. If Sarah is the eldest child it is likely that her parents married in the same parish. You could then ask to look up that marriage in a parish register.
Hope this helps,
Theresa