I should narrow the general scope of my offer a bit. Although I can lend general guidance throughout NB's various migration periods, I readily admit substantial difference exists between what my early Acadian "help" might be compared to say for Yorkshire Settlers, Planters, Loyalists, Home Children, Scot, Irish, German, Dutch migrations.
Given widespread "anglo-cizing" of names (Schwartz to Black, Stieff to Steeves, Leblanc to White) by many groups in Southeast NB, whether for gaining work, blending in, or avoiding expulsion orders and mortal dangers in the case of Acadians, the late 1700s/early 1800s is tricky at best for finding useful NB records. That said, there are 32 head of household census available free online for Southeastern NB from 1671 to the first full NB nominal census in 1851. All are accessible via 3 sites...
NB GenWeb Project http://sites.rootsweb.com/~cannb/
Mount Allison Univeristy https://www.mta.ca/marshland/index.htm
As an example, the "Census of the inhabitants in Dorchester in the year 1820" is on NB GenWeb and it includes a "Peter Cormia" (Cormier), likely family of the profile you raised in your post below, with household containing 1 man... 1 woman... 5 children.
Maritime Church records hold the real keys to early NB research though, but they are only now slowly shifting toward digitization, given the vastness of the archives and funding needs. There have been very recent advancements of projects though.
NB Genealogical Society just completed digitizing the NB Anglican Registers for their members.
Atlantic Baptist Archives (Acadia University) has several online datasets now.
United/Methodist Maritime Conference Archives (Sackville) remains fairly physical for their archives.
as does St. Michael’s Museum and Genealogical Centre (Miramichi).