Hi Blair

The arrival in 1834 at Castle Garden is on the IGI , showing the Vessel as the "Perdonnet" which arrived from Liverpool - which confirms most likely departure from the IOM. Most of the Passengers were Irish, Ann and family shown as English , not Manx and name Crane.Could not find death/burial on IOM 1830-1834. Possible death on the ship, very much happened in that time.
Could not find them for sure on US 1840 census, only two possibles - only head of the family listed.
Hamilton Ohio , Ann Crane , head age 30-40 , boy 5-10, boy 10-15 - where is Mary Ann ?
Found Hector Darling and family in Lafayette,Louisiana in1850, and in Hunt Texas 1860, cannot find them in 1870/1880 , looked at Canada in 1871 to no avail. Searching census for clues.
Could not find William and John for sure in the U.S.

Most likely parents on Mary Ann are John Craine and Ann Cannell M 14/8/1825 Braddan.Both of that parish. Children confusing possible :
Mary Ann 14/1/1827 St Georges Douglas, Fathers trade not given , mother given as Ann Cannon
William Richard 22/11/1829 St Georges , father's trade is Labourer and mother Ann Quayle. (There is also a William Robert born same year and he turns up in LeRoy OHIO with his parents in 1850.
John Frederick, St Georges 15/4/1832, father's trade a Carpenter , mother Ann Quayle.

That period does not show signatures or "making a mark", so if illiterate that could explain the range of surnames for Ann.

Possible births of John and Ann , which explains William and John.

Ann Cannell 23/7/1802 to John Cannell and Ann Creer, St Matthews Braddan

John Craine 24/6/1804 to William Craine and Jane Quay. Braddan.

Other posters may be able to confirm or dispute these suggeastions

Cheers
David Dwyer