You haven’t said what denomination your Morrow family was. That’s fairly important if you are going to search church records. Do you know? (I’d guess Church of Ireland or Presbyterian, judging by the main denominations in the county in the 1901 census).
To answer your question about where to look for records that are not on-line, the primary source will be PRONI in Belfast. Most records for what is now Northern Ireland have been centralized and are kept there, rather than in the individual counties.
You said that some of the family lived in Drumconor. That’s in Pomeroy parish. I checked the tithe applotment records for 1829, but there’s no Morrows in that townland then. So either they didn’t live there then or they weren’t farmers.
No Morrows in Drumconor in Griffiths Valuation of 1860, nor in the 1901 census (when there were 20 houses and a total population of 88).
You mention a William in Drumglass. Drumglass is the parish around Dungannon. Here’s the Williams farming there in 1832:
If you haven’t already done so, you might want to search this Co. Tyrone site for any mentions of your family:
PRONI has 3 documents you might want to look up if you are going there:
D2469/36 is the rent roll for Drumconor in 1786; D2469/51 is the same for 1805 and D2469/78 is same for 1826. If your family were farmers they may be listed there. If they were agricultural labourers/weavers they probably won’t be, as they tended to sublet from the farmer whose land they were on, rather than directly from the land owner.
For a good description of life in the area in the 1830s, you could read the Ordnance Survey memoirs. These were compiled on the instructions of the Duke of Wellington (then Prime Minister) primarily for taxation purposes. So a bit like the Doomsday Book. They were compiled parish by parish, and describe the inhabitants, their occupations, pastimes, habits, they analyse the various different denominations by number, and report on health, schooling, seasonal migration patterns as well as permanent migration patterns. And so on. A typical parish contains about 20 to 30 pages of information and some drawings. They are well worth reading if you want to get a feel for life there at that time. (It’s probably the most detailed contemporaneous summary that exists from that period).
There are copies on the bookshelves in PRONI’s main research room in Belfast. Omagh library (local studies section) may also have a copy. If not you can order a copy from the Ulster Historical Foundation. If you e-mail them with details of the parish(es) you are interested in, they’ll send you the relevant volume(s). Generally there are 3 or 4 adjacent parishes in each volume so if you are lucky you might only need to buy one or two. I think they are between £5 and £10 per volume depending whether it’s old stock or newer reprints.
Link to Omagh library heritage information:
The Mellon Migration library at the Ulster-American Folk Park contains a lot of genealogical books. You might find that worth a visit.
A word of caution. There aren’t that many records for people in Ireland in the early 1800s, so research for that period can be hard going.