I don't know if this is any consolation or not for you but I have a hunch that your H's LD problems go deeper than sex. For my LDH it definitely did/does...but don't despair, we're working through this and things are slowly improving.
What have you done (alone or as a couple) in the past to try to work through this? Counseling, Dr's, avoidance, patience, anything? I don't mean that to sound flip, I'm just curious what you've tried...for some people doing nothing is a course of action too.
As for me I did demand (well not really demand) but insist upon my H having a complete physical and all labwork. Now with him there was a process to getting him to agree to a Dr. First we had to try herbal remedies/supplements (which I knew weren't going to work, but what the heck.)...and of course I talked til I was blue in the face...which only served to push him away and become more resistent to me.
When an agreed amount of time went by and the herbal supplements didn't work (as expected) he did agree to go see his doc. I know this is a bit manipulative, but I told my H that a low testosterone level is sometimes a leading indicator to more serious physical problems (not a lie by the way, this is true). I just knew he wouldn't go to his doc just to have his T-levels checked. This approach worked with him...unfortunately (or should that be fortunately?) all came back normal.
Which led me to my next course of action (which by the way I knew we would end up at, and I suspect your H should as well...possibly you as a couple)....counseling.
My H was very reluctant to do...for him counseling had been a precursor to divorce in the past. He felt I was going to use the C to validate my reasons for leaving...until I told him and her (our C) that I didn't need her to validate any reasons I may have for leaving...if I wanted to do that, I'd do it. I was there to avoid that eventuality.
My H is continuing to see his C, and has one he is very comfortable with which is great! For us, the C has been a very important part of things.
Let me ask you this as well....has your H had a pattern of initiating in your relationship...or should I say a lack of it? My H NEVER initiated and I didn't even realize it until I really looked at our relationship closely. We (I really) have recently discovered why that is...which would never have come to light without the C
By the way...I've been in OK since 88'...I think by now I'm a native by duration. Originally from the Land of Enchantment!