Hi Cindy~ Sorry I don't remember the exact ages of your children, but no matter, they must be old enough to be concerned.
You are wise to concider their feelings. I also think that it can send a valuable message to them, to fight for marriage and to build a sense of what committment is.
Explain to them again that there are difficulties between mom and dad. That it does not mean you or dad love them any less. Tell them you cannot tell what the future will hold, but that you will always be there for them.
Explain your views on marriage and committment, in an age appropriate way. In keeping with that there are times that you and their dad will be together to see where things go. There will also be times that you will all be together. Be open and ask them to come to you with their feelings and questions.
DO NOT involve them in the specifics of the issues. That is between you and H.
I think the more up front you are with them the better.
Nothing wrong with being together as a family, then go home. Also, if it feels like you are making progress and staying the night will be ok. I wouldn't hide it from them, but maybe just not in their face kind of thing. Then that is ok too.
What if you were dating him for the very first time? When would you spend the night?