I found a manual for Sonic 3, albeit, the Hasbro version. Here's a link: https://instructions.hasbro.com/api/download/E9730_en-us_tiger-electronics-sonic-the-hedgehog-3-electronic-lcd-video-game.pdf
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Cool. Thanks! Yeah, it's the manual for the 2020 Hasbro re-release.
I did manage to find a scan of the original 1994 game at Sonic Retro. Here is the download link: https://info.sonicretro.org/images/e/e4/Sonic3_LCD_manual.pdf
I'll review some gameplay videos (like this one) to see if I can recreate the game, but I think I'll need to find someone who owns the original game to collaborate with for an accurate simulation.