Success! While I am not fully sure about what exactly went wrong, it seems that I just need to press quite a bit harder with this new propane cylinder (same brand, almost same age due to labeling.) That could be why the other tank ended up leaking, some defect making it easier for propane to come out (which would explain why I got used to pressing the propane tank more lightly. With a full fill, I got 63 full powered shots off, which really surprised me, as I was previously doing three 2 second fills, which gave it about 38 shots.
Thank you all, for you have helped me immensely!