I don't think it is unreasonable to ask him to curb his dating for a while, especially as your need for him is greater at this moment. If it was someone he had been seeing for years, then perhaps that is a different matter. The real issue that needs to be addressed is your insecurity, the fear that your illness will scare him and he will want to move on. To escape. I'm sure this happens in monogamous relationships too.
It may be the juxtaposition of his happiness/excitement and your sadness/fear that is triggering this. "How can he be out enjoying himself when this is happening?", which then leads you to question his commitment.
You need to talk to him about your fears. You said you don't want to burden him with problems, but that is exactly what he is there for. That's unconditional love. As difficult as it may be to communicate, and god knows I am pathetic at it myself, you need to express your worries and concerns.