I am not an authority either. I will have been married for only 4 months on October 5th & am also starting to realize that marriage isn't all that you think it's going to and "supposed to" be.
So, that having been said, I don't think your husband deserves you! You have obviously gone into this whole marriage with a commitment to be faithful, open, and honest and he, apparently, has not.
You have even asked for more information & for him to explain himself & make sense of very bad-sounding circumstances & he doesn't even sound like he's going to make the time to come up with a worthwhile response. He should not be talking with his ex-girlfriend or coming home at 5 am without a darn good valid excuse... but at least he should be "man-enough" to tell you the truth and not make excuses for being a jerk. If he doesn't care enough about you to even tell you the truth that's your first "deal breaker". Then, when you have to go investigate for yourself and he "changes the passwords"... that's like a signed confession that he has NO INTENTION of changing his ways and he does not care about your opinion of it &, therefore, DOES NOT CARE ENOUGH ABOUT YOU for you to be a part of this marriage.
Like I said, marriage is not what I expected either... my husband's cousin/best friend moved to our town (from out of state) & he is now spending all of his time with his cousin instead of me. Believe me, I have already had to contemplate the idea that our marriage may not work when I never had a doubt for the last 4 years of dating. It is REALLY scary, confusing and down-right awful to think that your marriage, wedding, everything was for nothing.... But then I think of my brother who has been married for 9 years to a woman who treats him like dirt every minute of every day... and he stays with her because he doesn't want to risk being a part-time dad to his son...
You need to do something drastic NOW!!! Either get out of the whole thing...OR, what I would probably do if you think he's worth it (and only you can make this decision) is tell him that either he goes to counseling with you or you're gone! If he doesn't think it's necessary, just keep repeating your offer : "either he goes to counseling with you or you're gone! " If you go and things are better or worse... decide from there... BUT, IF HE WON'T GO WITH YOU... you need to realize that YOU NEED TO DO WHAT'S BEST FOR YOU,,,, and if he doesn't care enough about you to try & fix your marriage... he doesn't deserve to be your husband.
I know I would be mortified to have to admit to my friends and family that the man I love cheated (or is cheating on me)... it is a MAJOR BLOW to you and your self-concept.... But to let it continue would eventually tear you apart. I was afraid to tell my friends and my family about my husband & worry that they "wouldn't like him or respect him anymore"... but they were really supportive & really helped me a lot. My mom did say a few harsh things about his actions...and you need to be prepared to hear those coming (even more so than I had)... but having people there to support you when you feel your worst really does help. Start with whoever you think has your best interest at heart- someone who would never purposefully hurt you... even if they have NEVER been in a similar situation or never-even been married. That's the person I told first & they gave me some really heart-felt answers that at least gave me some ideas of where to start.
Good luck & I wish you the best.
NewBride