Walls, I feel I'm getting this further off your topic, but these questions were asked, so I want to try to answer them. And the music does go toward showing a major difference between Primitive Baptists and
most other Baptists.
To Lone Pilgrim: Here are some tunes that I have heard used in Old Baptist churches. These are not all the tunes I hear in Old Baptist churches, just the ones to which I know the names. First is the page number in the Hymn Book, then the name of the tune, then a source where that tune may be found. I used the Broadman where I could, since it is a fairly common hymnal that many people have. I used the Harp of Ages where possible, since you said that you have a copy of that. The other references are from the Sacred Harp 1992 Revised Cooper Edition.
THE PRIMITIVE HYMNS BY BENJAMIN LLOYD:
3; New Britain; almost any hymnal in the South
5; Ninety-Third; The Sacred Harp, p. 31A
6; Devotion; The Sacred Harp, p. 48A
8; Ester; The Sacred Harp, p. 437
12; Heavenly Home; Broadman Hymnal, 261
16; Woodworth; Broadman, 162 - "Just As I Am"
75; Idumea; Harp of Ages, 32
96; New Britain - "Amazing Grace"
119; Columbus; The Sacred Harp, p. 67
121; Beach Spring; The Sacred Harp, p. 81; also to a tune similar to "The Fire Song - Oh, my dear brother, when the world's on fire, don't you want God's bosom to be your pillow..."
130; Sessions; The Sacred Harp, p. 38A
143; Duane Street; The Sacred Harp, p. 164
153; Cusseta; The Sacred Harp, p. 73A
155; Cusseta; The Sacred Harp, p. 73A
159; Lenox; The Sacred Harp, p. 40
184; I Would See Jesus; The Sacred Harp, p. 75
194; Forest; Harp of Ages, 110
196; Dennis; the very common tune used with "Blest Be the Tie"
201; King of Peace; Harp of Ages, 109
204; Weeping Sinners; Harp of Ages, 65
237; What a Friend We Have in Jesus; Broadman, 160
300; Brown; Harp of Ages, 163
322; She is Sleeping; The Sacred Harp, p. 540
329; Nettleton; Broadman, 190
330; Martyn; Broadman, 172
333; Mercy Seat; Harp of Ages, 12
338; Hinder Me Not; The Sacred Harp, p. 355
338; Arlington; Broadman, 176
345; Canaan's Land; The Sacred Harp, p. 101A
I have more but this is getting long, so I'll close for now. Generally, these songs are sung with only the melody - no alto, tenor, or bass parts. Sometimes that will vary, with a singer or a few adding a part. In regions where The Sacred Harp tradition is strong, the folks may sing their part from memory, and the church singing then sounds more like a Sacred Harp singing. To get the flavor of singing of Old Baptists here in the southwest (for churches that haven't
modernized their sound), here are a few pointers. Sing the songs much slower than you think they should be sung. Then add what musicians probably call "appoggiatura." These are melodic ornaments, like grace notes. In practice it is sliding up or down from one note to the next - usually hitting just above or below the "intended" note, then sliding to where you're "supposed to be." I don't really know how to explain it any better than that. It's best to just hear it.

Hope this helps.