As you mention, an S-step and then a 3-turn are part of the spin. You have identified issues with both of them, which will cause problems in whatever comes after them (the spin itself). I would suggest a 3-part approach: (1) 3-turns; (2) S-steps with 3-turns; (3) the spin itself. Below are details on each:
(1) 3-turns: See this article on “The Ultimate Three Turn,” along with the “Twin 3’s” exercise to practice. Make sure you always step on an edge, never a flat.
https://www.artofskating.org/the-ultimate-three-turn/
(2) S-steps with 3-turns: Practice ‘Alternating Forward ThreeTurns” from Moves in the Field. Make sure you bend your knee DEEPLY on the S-step (“step foward”). Here are two videos, neither is quite what you should be practicing. Oleg’s video has great knee bend and edge quality, but he’s using an open free leg; which is not wrong, and maybe something to practice for your spin entry, but you should practice first with closed free leg. Also, I don’t like how far apart Oleg’s feet are on the S-step. The other video has closer footwork, but not enough knee bend to really step onto a clean FO edge on the S-step.
https://www.youtube.com/watch?v=GoY8Haa1Ee8&t=23s
https://www.youtube.com/watch?v=GoY8Haa1Ee8
(c) The Spin itself: Go to YouTube, type “Gary Beacom Spin” into it, and watch every relevant video. Here is one video where he talks about the importance of a deep entry edge on the crossovers, then a deep S-step into the spin. It’s sit spin here, but that applies to you as well:
https://www.youtube.com/watch?v=L_PhicNu1Lc
Good luck!