OP, your H is also addicted. If he wasn’t then not drinking wouldn’t be any issue. He’s coming out with all the excuses I’d expect from someone who is dependant on drink, and I’ve had lots of experience.
Either way, giving up drink is all on you, there’s no way around it. Even if it doesn’t enter the house, it’s everywhere, so there will never be a day when you don’t encounter it a hundred times.
Rather than lie, I suggest you be honest with your H. His reaction will tell you all you need to know and then you can decide how to proceed from there.
Good luck. I know how hard it is for someone to give up the booze when it’s become such a huge part of their life. My ex carried on even though he was by all measures a complete wreck because of it. I don’t think he’s even stopped now, and it’s cost him his job, his son, his mobility, his eyesight and god knows what else.
Take it one day at a time and just focus on getting through the next evening without. Is there something else you can do to take your mind off it? Read a book, take the dog for a walk, take up knitting, just something to distract yourself and give your brain something else to concentrate on.
Good luck again. You can do it, and despite how it looks now, there is life after booze.