I think that when you're married that you share financial responsibility and that you DO have a right to know what's going on, especially if it gets so bad it makes it hard to make the payments you share together, like your house, etc... You should definitely be able to look in every part of your house if you're trying to make sure that there isn't anything illegal in there since you would be liable as well as him.
It sucks that you feel put in that position in the first place though and that you don't feel you can trust him, which makes it heavier on your heart. From my experience, when you can't trust them with one thing it broadens the scope of what else they could be lying about. When I want to get back with my ex I try to think about all of the lies he told me and then double it when I think about how many other lies he probably told me that I never knew about.