Bro. James, I noticed that Daniel hasn't been around in a few days, and think I'll take on an attempt to answer your question. I can only speak as an outsider who has read about the Old Regulars, corresponded with a few elders, and listened to some of their singing. So if an Old Regular Baptist online can add to or correct this, I'll welcome it.
In my experience, all Old Regular Baptist singing is slow and lined out. From musical history it appears that part of the reason for this was that when the singing school movement was becoming popular, they rejected the "regular way" of singing (which is basically our shape note singing with note books) to hold on to the "traditional way" of singing (lined-out hymnody from words-only hymn books).
Here is an article kind of related to the subject:
The Regular Singing Controversy
I think this one contains some info also:
Psalmody & Church Music
I would add that probably most Old Regulars believe that using books with tunes takes away from the spirit and the Spirit in the singing. I think you will find also that Primitive Baptist singing in the same general areas as the Old Regulars generally tends to sound more like the Old Regular singing than Primitive Baptist singing in Texas.