And FWIW, I would ask for passwords to all of his online accounts including his mobile phone bill. That way you can check whenever you want so he doesn't have a chance to go through and delete or change things. You will probably obssess about checking at first but eventually, if you don't find anything suspicious, you'll stop checking all the time.
If he has nothing to hide then he won't have a problem with it. I'm not sure how it would work since your H is a L and client confidentiality is an issue. But he should be willing to figure out a way to make it work somehow.
IRT stop thinking about OW: yes, that's ideal. But we're only human and the specter of the OW will haunt you for a while. Eventually you get to the point where you realize that there's always a way to be sneaky and if H really wants to see/talk to/be with OW then he will. But at the beginning it's completely understandable and reasonable that you want to be reassured and you want to verify that he is being honest. As I explained to BF, he's proven himself to be a liar and he now needs to prove himself to be honest.
If you love somebody, set them free. http://tinyurl.com/2empx2g