Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/15/63, followed by the parenthetical note: Thus Kreisel saw an analogy between forcing and Friedberg's  priority argument. The following page suggests other logicians agree that forcing was implicit in priority arguments from recursion theory (p. 158) and mentions Kunen (who is cited in one of the answers here). On the same page, it is mentioned that Kreisel claimed he had a form of forcing in his interpretation of intuitionism in the 1961 paper "Set-theoretic problems suggested by the notion of potential infinity." However, Moore contends that Cohen was the first to use forcing and related ideas in Set Theory. Nevertheless, I am grateful for the responses thus far, and would warmly welcome further clarification regarding my question below from other experts in the area of Set Theory and/or Mathematical Logic.
Background: Paul Cohen began to think deeply about the Continuum Hypothesis in 1962, and published his proof of its independence (in two parts) by the following year. Of course, there were earlier mathematical moments that led to his "discovery of forcing," including his familiarity with Skolem's work (in particular, the Löwenheim–Skolem theorem) and a desire to think in terms of "decision procedures." I will include a few relevant references at the end of this question, including a retrospective/introspective piece by Cohen himself.
My question concerns an occurrence in 1957, when Richard Friedberg provided a solution to Post's Problem. First, allow me to transcribe an excerpt from Cohen's talk at the 2006 Gödel centennial:
At that time there was great interest among Raymond [Smullyan] and some other people about the Post Problem. And that’s a problem which could have interested me; it had a mathematical flavor to it. But I never thought about it, and occasionally we’d have coffee and I’d hear these people talk about it. But one day, someone came to my office and said, “This problem’s been solved.” And I said, “Really?” “Yes, here’s the letter. I can’t believe it’s true!” And he gave it to me and I read it. I went to the blackboard, took some chalk, and I said, “Well, it seems right.” This is the proof by Friedberg – and so that was my only contact with logic at that point. But I still never lost this idea of somehow thinking about the foundations of mathematics: trying to find some kind of inductive technique for simplifying propositions; perhaps leading to a decision procedure, when impossible.
This talk is summed up in the introduction to a re-printing of "Set Theory and the Continuum Hypothesis" (Cohen, 2008) in which the remarks corresponding to the above excerpt are:
A small group of students were very interested in Emil Post's problem about maximal degree of unsolvability. I did dally with the thought of working on it, but in the end did not. Suddenly, one day a letter arrived containing a sketch of the solution by Richard Friedberg (Friedberg, 1957), and it was brought to my office. Amidst a certain degree of skepticism, I checked the proof and could find nothing wrong. It was exactly the kind of thing I would like to have done. I mentally resolved that I would not let an opportunity like that pass me again.
I find the last sentence of this latter quotation rather interesting, particularly since it concerns a time five years before Cohen's work on ~CH officially commenced. A quick check of Wikipedia gives a problem statement and solution (i.e., the priority method) that sound remarkably similar, at least on the superficial level, to Cohen's subsequent work with forcing. Unfortunately, work on Turing degrees falls well outside of my bailiwick.
Question: Can someone who specializes in Set Theory or Mathematical Logic comment on the similarities between Post's problem/the priority method and ~CH/Cohen's forcing? In particular, is there reason to believe that what was Cohen's "only contact with logic" by 1957 would have contributed in a meaningful (mathematical) way to his work half a decade later?
Cohen, P. (2002). The discovery of forcing. Rocky Mountain Journal of Mathematics, 32(4).
Kanamori, A. (2008). Cohen and set theory. The Bulletin of Symbolic Logic, 351-378.
Moore, G. H. (1988). The origins of forcing, Logic Colloquium ’86. Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 143-173.