"Out of the Depths" is essentially a modernization of John Newton's own autobiography, "An Authentic Narrative," by which I mean it updates the language to 20th century American English, organizes it into book chapters (Newton originally penned his autobiography in a series of letters), and so on.
However, if you want Newton's "An Authentic Narrative" as he originally wrote it, it'd be best to purchase "The Life and Spirituality of John Newton" by Bruce Hindmarsh (ed.). "The Life and Spirituality of John Newton" contains "An Authentic Narrative."
To better see what I mean, here's the first paragraph of the first letter in "An Authentic Narrative" followed by the first paragraph of the first chapter in "Out of the Depths":
* * *
Reverend and Dear Sir,
I make no doubt but you have at times had pleasing reflections upon that promise made to the Israelites, Deut. viii. 2. They were then in the wilderness, surrounded with difficulties, which were greatly aggravated by their own distrust and perverseness: they had experienced a variety of dispensations, the design of which they could not as yet understand; they frequently lost sight of God's gracious purposes in their favour, and were much discouraged by reason of the way. To compose and animate their minds, Moses here suggests to them, that there was a future happy time drawing near, when their journey and warfare should be finished; that they should soon be put in possession of the promised land, and have rest from all their fears and troubles; and then it would give them pleasure to look back upon what they now found so uneasy to bear: "Thou shalt remember all the way, by which the Lord thy God led thee through this wilderness."
* * *
God's promise to the Israelites is worthy of our reflection. They were then in the wilderness, surrounded with difficulties that were made worse by their distrust and disobedience. They had experienced a variety of God's providential actions, but they did not yet understand God's ultimate goals. They frequently lost sight of God's gracious actions on their behalf and were very discouraged. Moses announces to them that there is a future time of happiness drawing near when their journey and warfare will be finished. They will soon take possession of the Promised Land and rest from all their fears and troubles. Then they will look back with satisfaction upon what was before so difficult to bear: "Thou shalt remember all the way which the Lord thy God led thee through the wilderness" (Deut. 8:2).