Sorry it's long... I met a man, we had great chemistry on all levels, he seemed (and acted, and sounded) very attracted, and very interested in me and in a relationship, but physical contact never went beyond hugging and kissing (deep, erotic). Kissing always took place before we parted, never at the beginning or during the date/meeting.
I must add that during the month we were dating, he had one "attack" of hesitation and not being sure he wanted to continue, but he changed his mind the next day, and came back saying he had no more hesitations.
At one point he invited me over on Saturday morning, to spend the day. He even introduced me to one of his daughters. During that day again, everything seemed great (he had baked and cooked for me), but we never went beyond those kisses... at one point, when I was laying on the sofa, he lay down beside me and hugged me, but a few seconds after I put my arm around him, he moved it away gently, and moved away from the sofa.
The next day he dumped me, saying the following confusing sentence: "this is getting serious, but I don't think it can be serious".

I have a very strong feeling that if physical intimacy was not the problem, it was definitely a big part of whatever it was that caused him to break it off. I do realize it could have been something even bigger.
A week later, he sent me an email, asking how I was. He referred to me by my nickname (the one he gave me), and signed with his (the one I gave him). He has been emailing short messages and sending SMS messages every once in a while (it has been a month since he broke it off). He obviously wants some kind of contact with me, and he obviously feels more comfortable in a friendship situation, but it's not what I want.
And my question is: should I allow friendship, in hope that something will change? Or should I cut it off completely (in the hope that something will change...) Is there anything at all I can do, or should I just put it behind me? (needless to say, I really want this guy...)
Obviously, my questions are tactical... I know there are other important questions to be asked, such as why bother with someone who dumped me, and I ask myself these questions every day, but still, please answer the questions I have asked here.
BTW, he is 50 yo (I'm 45), has been divorced for 6 years, has 3 kids, and until a year ago was in a relationship that lasted 3 years (but I have no idea what that was like, or why it ended. I only know they didn't live together.)

Thanks in advance!