Few Irish Parish registers go back far enough to include birth year like 1806. I think only two parishes of 41 in Sligo are that early. If he travelled alone or if his parents stayed behind you might find them in the Tithe records of the 1820's another census substitute like Griffith Valuation.
John Henry had no son's? as likely he would of named his 1st son after his father (or himself)
Who is the Adam's family - intermarriage?
Are not the relationships shown to the Henry's, boarder, in-laws etc. Does Sara J. say married sister? or with the ? are you assuming an association.
Sometimes even though documents don't show anything other
than Ireland, passenger records might show where they came from (other than a port of departure) Gravestone inscriptions often refer to the County of birth!
Co. Derry had more Henry's than Co. Sligo 265 to 160 in Griffith Valuations of the 1850's. 23 counties have both the surname Henry and Adams. Pretty much all.