I appreciate that you admit you have some jealousy issues because of past situations, but his behavior still sounds a bit .... off, to me. Now, I will say that in my last relationship, it was him who was slightly jealous, but we got past it by keeping things completely open. Meaning, if he thought I was behaving at all strangely, he stated it flat out, right away and gave me a chance to respond. But, I will also say that I at first had a slight tendency to "minimize" things, when I knew nothing "bad" was happening, but just didn't want to upset him needlessly. Over time, I realized that did me know good and our communications cleared up and the jealousy cleared up.
Unfortunately, our roles were switched when he had an emotional affair years later, and he did behave in some of the ways you are mentioning about your DH. So I'd say, be straight forward, ask WHY he may feel uncomfortable talking about Betty with you, ask if there are any issues that he feels need to be worked on that you may be unaware of, state your concerns, in a calm, non-attacking way.
Good luck. You have every right to have suspicions, but he has every right to have a chance to explain himself, etc.