Julie -
I think your emotions are totally normal. You are in love with someone and you want to spend time with them and you want them to want to spend time with you. There is nothing to feel bad about.
However, because he is sick and actively trying to get better, it is going to be hard for him to attend to you and address how you feel. You must accept that it will not be a "normal" relationship in the way that you want it to be. In the future when he has been sober for some time, and his mind and body are healthier, he may then begin to address concerns within the relationship. But, I don't think that is the main concern. Like you have mentioned, you're not so much worried about his recovery as you are about whether or not he actually likes you.
And the truth is, you won't know until he is healthier. He may have only entered the relationship out of need or loneliness. There is no doubt about it that you are taking a risk. It is up to you whether you feel the risk is worth it. In order to quiet your doubts you may try to get assurance from him - which is why you feel inclined to call. But you must realize that this is unfair. He cannot give you the assurance you need.