Yes, you should be able to get that achievement so long as you complete all the puzzles, the order/playthrough doesn't matter and it should still work on skips.
Ah, that's good to know! Thank you for your reply! Really enjoying the game, btw. You really know how to tell a good story! Gotten most of the endings now. Currently working on solving all the puzzles (on my own) before I go for the remaining endings.
