Your insecurities are something that you need to work on. Do not ask your romantic partners to help you with that, because it is not their issue and it will only serve to hurt the relationship. You need to remind yourself of your self-worth, however you go about doing it. You were just in a 6 years relationship and things are going well with the new guy, so obviously you don't have any problem attracting guys. If you understand that your insecurity is not based on reality, but on your own internal conversations, then it might help you deal with it better.
As for the new guy, I applaud you two for your mature understanding of what a relationship actually is and not wanting to label and define a relationship that hasn't even started yet.
Good luck.
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