Well, William Dixon is a very common name, so I'm not sure that it makes any sense to pick any particular parents at this point. If you have complete lists of children, then you may be able to find who their grandparents are.or you might be able to find a marriage record which lines up with grandparents. My opinion is that both sets of parents should remain intact and look for as many sets of data as can be found... I was looking in Ancestry.com and there seem to be at least two William Dixons in England an at least one in the USA.
Probably both sets of William / Parents should have a bunch of Research Notes with links to make it easier to sort through the data.
BTW, I have Dixons who are ancesters of my cousin's but I don't remember how far back I've traced them. I believe they are from WV and later moved to Ohio.