sounds like you're doing well. When H sends a text or leaves a message, how much time do you give it before responding? Just curious and wonder if you might want to delay your responses further to demonstrate that you are busy and getting along OK.
It sounds to me like he doesn't want to disconnect. If he did, he wouldn't be contacting you daily. And because he hasn't shown this degree of concern for other friends tells you that this sitch is different. You have to decide how to handle the contacts because I know you don't want to push him away but you want him to act like your H not just a friend.
Food for thought... by building a strong friendship with your H you may lay the groundwork for more at a later time. Also, are you sure he's just checking in to make himself feel like he's really there for you, or is that your assumption? If so, maybe there are other reasons for checking in that you don't yet understand? Just something to think about...