I've looked up a few Co. Tyrone Hazletts for a collateral cousin, but none that exactly match your ancestors. Maybe some of these notes and suggestions will help you, though.
Location is all-important in Irish research, so identifying your Co. Tyrone ancestors with particular places is vital.
"Donemanna " (Dunamanagh or Donemana), the district where several of the Hazlett children's births were registered in the 1860s-1870s, is a townland and town in the civil parish of Donaghedy (Donagheady), in the barony of Lower Strabane, Co. Tyrone. The parish is on the northernmost border of Co. Tyrone with counties Donegal and Londonderry. I'd definitely suggest researching Kanes and Hazletts in those counties as well as in Tyrone.
Tithe Applotments books (1820s-1830s) for Donaghedy parish have not survived. In other parishes, however, tithe-payers in Termonamongan parish in 1828 included Daniel KANE of Croighdennis and an unknown KANE (perhaps an illegible name) of Ballymungan. Also in Termonamongan parish were Cooper HAZLETT and Thomas HAZLETT of Leitrim (Caldwell).
Termonamongan parish is on the far northwest side of Co. Tyrone and doesn't seem close enough to Donagheady parish for a man and woman to have met and married (when courting distance was generally as far as your legs or a horse-and-cart could carry you on an afternoon) .... but it's not impossible.
Griffith's Valuation (completed 1859-1860) shows a "Daniel KANE" farming in Termonamongan parish, and a man of the same name in Derryloran parish. HASLETTS in Donaghedy parish in Griffith's were:
-- Hugh Haslett and James Haslett at Dullerton (note that your ancestor Andrew may have been living with and sharing tenancy with a brother, his father, uncle or other relative, and not counted on his own. His father Jacob Haslett may have died, leaving a son as main tenant of the family farm)
-- John Haslett at Stonyfalls
-- Isabella Haslett at Magheramason
-- Edward Haslett at Rousky
"Andrew HAZLETT" was farming at Castlegore, Urney parish, Co. Tyrone.
In Termonamongan parish were:
-- Hugh Hazlett at Carnoughter
-- Robert Hazlett at Carndreen
-- John Hazlett atFirst Corgary
If you haven't checked further on these men and other Kanes and Hazletts counted in the Tithes and Griffith's Valuation, I'd recommend doing that.
This excellent Co. Tyrone site -- http://cotyroneireland.comhttp://cotyroneireland.com -- has a database of 1st Donagheady Presbyterian Church Marriages, 1845-1928.
21 Jan 1846 - Eliza J. Haslett of Dullerton, father: Jacob Haslett, Farmer, to James Thompson of Altrest, father: Andrew Thompson, Labourer.
11 March 1850 - Isabella Haslett of Dullerton, father: Jacob Haslett, Farmer, to Thomas McIntosh of Terkenaghan, Farmer, father: Thomas McIntosh, Farmer.
20 Feb 1877 - Rebecca Hazlett of Dullerton, father: John Hazlett, Farmer, to John Nutt of Gortnaskey, father: David Nutt, Farmer.
18 Oct 1877 - Nancy Hazlett of Dullerton, father: John Hazlett, Farmer, to Thomas Curry of Craigtown, father: Robert Curry, Farmer.
13 Nov 1877 - Hugh Hazlett of Dullerton, Farmer, father: John Hazlett, Farmer, to Margaret Jane Curry of Craigtown, father: Robert Curry, Farmer.
So now you know two of Andrew Hazlett's sisters: Eliza J. and Isabella. And I think John Haslett of Dullerton was a brother.
In 1st Donagheady Presbyterian Church's Financial Statement of 1857 (also on http://cotyroneireland.com)http://cotyroneireland.com), pew-holders that year included James, John and Andrew Hazlett. I'd say that these 3 men were very likely related, so you probably have two of Andrew's siblings, uncles or first cousins there.
Further research in First Presbyterian church records might give you more on all these families. Donagheady Presbyterian Baptism records, for instance, begin in 1838. You'll probably find more of Andrew and Martha's children registered there. Andrew Hazlett's marriage to Martha Kane would have been in her church, which may have been another church and not one in Donagheady parish, so you'll have to research further to identify what church Martha belonged to.
Grange Burial Ground in Donagheady Parish includes a Hazlett grave (see http://tammymitchell.com/cotyrone)http://tammymitchell.com/cotyrone): "Here Lyeth the Body of Martha Hazlett Who Departed this Life Dec. 19th 1802 Aged 25 Years."
Since Presbyterian families often registered vital events in the official Church of Ireland, I'd check St. Michael's Church of Ireland records as well as Presbyterian churches of Donagheady parish. See PRONI's church guides here: www.proni.gov.uk/guide_to_church_records.pdf and others on various Co. Tyrone web sites. Sometimes the "surviving" or "available records" dates listed in these guides may not be correct; check all possible sources for church records you need.
I'd also suggest checking the 1901 Ireland census for Hazletts and Kanes. Hugh Hazlett, age 54, was living at Dullerton in 1901 with wife and children. Surely he's the "Hugh" of the 1877 marriage. Are there any descendants of this family in that area today? Find out and contact them!
To help others who may be researching this family, New York ship passenger lists show the Andrew Hazlett family arriving from Ireland in 1875:
Andrew Haslett - 65 - Labourer
Martha - 45
Mary - 13
Andrew - 11
Bella - 9
Joseph - 7
Youngest daughter Martha, born 1874, must have died, since a child that small would surely be in her mother's care and not left behind in Ireland. I think you might have daughter Mary's estimated birth year (1869) a bit later than she was actually born (c. 1862, by ship record). She wasn't with her parents in the 1880 census, but perhaps she was working in another household at the time or had already married her first husband.
Andrew Hazlett (age 70) and wife Martha (age 57) are in the 1880 U.S. census of Cambridge, Massachusetts. The only children with them at that time are Isabelle, 14, and Joseph, 12.
In 1900, Martha Haslett was still in Cambridge, widowed, age 74, born Feb. 1826, mother of 11 children with 5 living.
Since Martha Kane was a decade or more younger than her husband Andrew Hazlett, and she had children at least from c. 1850 until 1874, I'd say they married no earlier about 1845 (and had children for 30 years). Keep in mind that her surname might have been spelled Keane, Kaine, McKane, O'Kane, etc.
Since you've identified the Hazletts/Hasletts positively with Donagheady parish in Strabane, I'd recheck LDS FamilySearch Irish records for all people of that surname in the Strabane district: births, marriages, deaths (not just specific children of Andrew and Martha)
Are you related to, or in touch with, Ancestry researcher and tree-submitter "tchazlett" who lives in Cambridge, Massachusetts? He's researching the Hazlett, Garside, Aulie and Radcliffe families.