The song appears to be a late version of "Gideon's Band" (1861) I have it as arranged by C. R. Dodworth, but I'm not sure of the composer. The chorus is almost the same:

If you belong to Gideon's Band
Then here's my heart and here's my hand,
If you belong to Gideon's Band
We're hunting for a home.

The original verses are different though:

Oh keep your hat upon your head (3)
For you will want it when you're dead.

Oh keep your nose upon your face ... etc.

The verses you give are common to a number of "Noah's Ark" songs, such as "One More River to Cross," "Noah's Ark Shanty (A Long Time Ago),one of the two songs known as "Noah's Ark," etc. They were spread about by innumerable collegiate singing groups in the late 19th and early 20th centuries.

One band that recorded "Gideon's Band" with the "Noah he did build the ark" verses was Al Hopkins and His Buckle Busters in December 1928. However their verses are somewhat different from yours.

By golly, though, that Barnegat in your final verse is unique as far as I know. A New Jersey coastal version?

Bob