Kori,
I’d be inclined to disregard the Longford baptism you found. Andrew clearly said he was born in Tyrone so I’d rely on that. Longford is over 100 miles south of Tyrone. Not likely to be the same family. Many church records from that period have been lost and of those that still exist, many are not on-line. Only the RC records are routinely on-line and even there, there are some parishes that are not. I am guessing that you have searched Irish baptism records on Familysearch and just found that one Andrew in Longford. That doesn’t mean it’s the right one.
You haven’t said what denomination Andrew was but I suspect Church of Ireland. So they may be the records to focus on. Here’s a map of all the 43 parishes in Tyrone. You would need to work through them all in PRONI. (Not all have records for 1811. If he comes from one of those parishes whose records are lost, you will struggle to find the family).
https://www.ancestryireland.com/civil-parish-maps-for-ulster/civil-pari…
Families often repeated names from generation to generation. Obviously your Andrew had left Ireland long before 1901 but it’s possible that related family were still using the name. I searched the 1901 census for Andrew Fraser, Frasier and Frazer in Tyrone – none. However I did find one Andrew Frazier:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Drumharvey/Make…
In the 1911 census it’s spelled Frazer. (Expect the spelling to vary. People were not consistent with spelling then).
http://www.census.nationalarchives.ie/pages/1911/Tyrone/Drumharvey/Make…
This is probably his birth (Andrew Thompson Frazier) in 1894, in Tullynincrin, to Andrew Frazier and Leah Coulter:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_retur…
Andrew, the father died in 1894 aged 35. Leah remarried in 1896 to Irvine Stewart.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
So the Andrew, described as a boarder in the 1901 census is actually Leah’s own son!
Makenny is in the parish of Kilskeery. Their Church of Ireland records start in 1767. This is just a hint that perhaps your Andrew came from that area. At least it’s in the right county. Kilskeery baptism and marriages are on-line on this site. I don’t see any Frazer (or variant) events in them.
http://www.igp-web.com/IGPArchives/ire/tyrone/churches/kilskeery-mar.htm
http://www.igp-web.com/IGPArchives/ire/tyrone/churches/kilskeery-bap.htm
All I can suggest is you search other nearby parishes in Tyrone. Most are not on-line and so you need to go to PRONI in Belfast to look them up.