The problem is that H. is so unwilling to come back for a trial period, or work on our marriage at all, that I feel my hands are tied.

You have to go about things not contingent on H's coming back or not. That will untie your hands. Assume he's not coming back, as that more accurately reflects your reality, and look at your plans for life as if he will not be a part of it anymore. There's nothing about that, in the event of reconciliation, that would be an obstacle, not even the selling of the house or being divorced.

I have been told that even if we do D., H. would have to continue to cover my medical as part of the settlement. So, my health insurance isn't something I've been worried about. I hope I haven't been misinformed... I am trying to read up on divorce laws in my state

You're talking about the COBRA law. More accurately, in the event of divorce, you're no longer on H's policy, and the insurance provider has to offer you the same or comparable coverage you were getting while married, for a period of six months. However, it becomes your financial responsibility, and may cost more than before. After six months, the insurance stops. It's provided so that you're not left in a lurch and have time (six months) to make arrangements for yourself.

Now, in a SEPARATION settlement, you can have H continue to provide and pay for your insurance, as in a separation you two are still legally married.

Even in a DIVORCE settlement, you could have H agree to continue paying your premiums, but that would have to be negotiated.

NY has a type of "no fault" divorce available in two ways. You can file a separation agreement with the court and be legally separated for a period of one year, after which either party can file for divorce on the grounds of 'legally separated for one year'. That is not expensive to do, and it gives you one year's time if you're in no rush.

The second way is to file for divorce on the grounds of 'constructive abandonment', which simply means that one party (you) has been abandoned by your spouse for a period of one year. The other party signs off neither admitting nor denying anything, but agreeing to the divorce. Nice, clean and simple.

Any other grounds will cost you big bucks and lots more emotional grief. If you think of suing, for example, on grounds of adultery, you're going to have to go trial, and the associated court costs and legal fees, and the cost of proving the charges to the satisfaction of the court (PI videos/witnesses, etc,), will FAR exceed what you'd pay for obtaining the same divorce on either of the two above-mentioned "no fault" type grounds, where only filing fees are involved (and minimal attorney fees).

I still do not want a D. and I feel that it would be a mistake to do that. Maybe my H. needs to hear me tell him that I love him, that I forgive him for what has happened and that I believe in us, despite what took place. This is a man who has told me he hates himself; he can't look in the mirror at himself because of how ashamed he is. He is not making himself happier. At this point I think I need to be honest about my feelings for him so he understands that it is possible for me to love him despite what he did. Maybe he can take that away with him and think on it a while.

You can try that, but sweetie, you need to look in the mirror and ask yourself, "But why???" This is also a man with a big problem and not a good partner for a relationship.