I'm so sorry that you're in this situation. It is truly horrible, but if you aren't willing to do whatever it takes to find out what is happening to your life, you're at his mercy. My WH had been lying to me for so long, he didn't even know what the truth looked like. He NEVER would have voluntarily come clean with me.
This is NOT being nosy or sneaky! This is your life! He is withholding the truth about YOUR life from you, your marital life, your sexual life, your financial life, your health, everything!
By lying, he is exposing you to financial insecurity. He is endangering your life if he is engaging in unsafe sexual behavior. He is endangering your children's lives and security.
Getting the truth that he is deliberately hiding from you is protecting yourself and your children! He has broken the promises he made to protect you and them. Now, it's your job to protect all of you.
Please, use the time you have to go through everything you possibly can. Get beyond your sense that snooping is somehow betraying your values. He is the one who is betraying you and your children. You have to get the truth for yourself because he is deliberately hiding the truth of YOUR life from you.
If you insist on playing by rules that he is not bound by, you are helping him deceive you. Take some control back for yourself and your children. Yes, this doesn't feel like you. Being betrayed changes us. Your decision now is whether or not trusting him to make good decisions for all of you is working or not. Knowledge is power, and right now, he's the only one who knows what's really going on.
Take your power back!