Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search through.
Regarding the papers listed below, I looked through a lot of papers that I've collected over the years on elementary logic issues (far more than what’s below) and I’ve singled out some papers that might be worth looking at. I’ve divided these papers into two categories: (A) $7$ papers in which I saw some explicit connection with high school algebra topics being made (identified by italic bold numbers --- [9], [16], [18], [19], [21], [23], [29]), and (B) the remaining $22$ papers, which seem to have something of possible pedagogical interest for what you’re dealing with, even though they do not appear to make any explicit connection with high school algebra topics. I googled the titles to find the digital versions I’ve given links to, but nearly all of them are behind paywalls. There might be some papers that I’ve given a paywall link to that can be found elsewhere not behind a paywall, but I did not put any effort into looking for this.
[1] Oliver D. Anderson, How many Venn diagrams are there?, International Journal of Mathematical Education in Science and Technology 19 #2 (1988), 299-305.
[2] Oliver D. Anderson, A universal Venn diagram, Mathematics and Computer Education 22 #2 (Spring 1988), 78-80.
[3] Keith Austin, Set algebra: word proofs and Venn diagrams, International Journal of Mathematical Education in Science and Technology 18 #3 (1987), 482-484.
[4] Christopher Baltus, A truth table on the island of truthtellers and liars, Mathematics Teacher 94 #9 (December 2001), 730-732.
[5] Margaret E. Baron, A note on the historical development of logic diagrams: Leibniz, Euler and Venn, Mathematical Gazette 53 #383 (May 1969), 113-125.
[6] John D. Baum, An arithmetic method in symbolic logic, Mathematical Gazette 56 #396 (May 1972), 91-95.
[7] Javad Behboodian, Set identities and set equations, International Journal of Mathematical Education in Science and Technology 22 #1 (1991), 123-126.
[8] Max Black, The relevance of mathematical philosophy to the teaching of mathematics, Mathematical Gazette 22 #249 (May 1938), 149-163.
[9] Brother T. Brendan, A popular fallacy, School Science and Mathematics 59 #7 (October 1959), 509-513.
[10] Arthur H. Copeland, The algebra of logic, Pi Mu Epsilon Journal 2 #7 (Fall 1957), 317-323.
[11] Carmine DeSanto, A classroom note on Venn diagrams for four and five sets, Mathematics and Computer Education 18 #2 (Spring 1984), 107-110.
[12] Melvin Fitting, Propositional logic using elementary algebra, Mathematics and Computer Education 16 #3 (Fall 1982), 204-207. [related to Goodstein’s paper]
[13] Reuben Louis Goodstein, Solving equations in the algebra of classes, Mathematical Spectrum 2 #1 (1969-1970), 25-28. [related to Fitting’s paper]
[14] David W. Henderson, Venn diagrams for more than four classes, American Mathematical Monthly 70 #4 (April 1963), 424-426.
[15] Clive Kelly, Set algebra via Boolean functions, International Journal of Mathematical Education in Science and Technology 20 #3 (1989), 478-480.
[16] Ernest Albert Kuehls, The truth-value of $\{\forall, \; \exists, \; P(x,y)\}:$ a graphical approach, Mathematics Magazine 43 #5 (November 1970), 260-261.
[17] B. Meltzer, Mathematics, logic and undecidability, Mathematics Gazette 51 #375 (February 1967), 16-25. [See Corrigendum in MG 51 #376, May 1967, p. 134.]
[18] Karl Menger, On necessary and on sufficient conditions in elementary mathematics, School Science and Mathematics 39 #7 (October 1939), 631-642.
[19] Kenneth A. Retzer, Logic, tradition, truth in algebra, and inferences for geometry, School Science and Mathematics 84 #3 (March 1984), 181-188.
[20] Kenneth A. Retzer, Proofs with visible inference schemes, School Science and Mathematics 84 #5 (May-June 1984), 367-376.
[21] Joseph V. Roberti, The indirect method, Mathematics Teacher 80 #1 (January 1987), 41-43.
[22] Mariano Rodrigues, Coding Venn diagrams, Mathematics and Computer Education 15 #1 (Winter 1981), 46-49.
[23] Myron Frederick Rosskopf and Robert M. Exner, Some concepts of logic and their application in elementary mathematics, Mathematics Teacher 48 #5 (May 1955), 290-298.
[24] M. Stephanie, Venn diagrams, Mathematics Teacher 56 #2 (February 1963), 98-101.
[25] Ian Stewart, The truth about Venn diagrams, Mathematical Gazette 60 #411 (March 1976), 47-54.
[26] Charles W. Tryon, The logic of implication, School Science and Mathematics 73 #1 (January 1973), 56-60.
[27] C. H. Wild, The "characteristic function" and set theory, Mathematical Gazette 54 #389 (October 1970), 296-297.
[28] Margaret Wiscamb, Graphing true-false statements, Mathematics Teacher 62 #7 (November 1969), 553-556.
[29] John D. Wiseman, Complex contrapositives, Mathematics Teacher 58 #4 (April 1965), 323-326.