"Happy in the Prison" has a strange mondegreen. The line should be "Pentecostal Rain is falling," not 'reign'. The subject of Latter Rain and the Latter Rain Movement and Latter Rain Covenant in the Pentecostal creeds was a subject of much discussion, ca. 1910. If the Holiness Quartet has 'reign,' they also are guilty.

It has to do with the prophet Joel, Old Testament, and the 'latter rain' outpouring of the Holy Spirit in the ends of time.
It may be that, latterly, the subject has been forgotten by current pentecostals. Googling will bring up the information on the meaning; I am not that interested in the peculiarities of the pentecostals.

I doubt that the Af-Am spiritual is the source for this piece of white gospel; laying down burdens or transferring them to Christ is a part of several creeds and it is in their hymns. See "Laying the Burden Down, (Romans and Matthew)," a Presbyterian talk- Laying the Burden Down

I have the same doubt about a couple of other songs that you link to Af-Am spirituals which more likely are based on white gospel. A difficult subject; there are many published white gospel collections, some privately published, locally, and in very small numbers- it takes a specialist which I am not.