Hm. I met a guy of secure attachment style online. His personality was great. He did have some insecurities but ultimately you could tell he was a very healthy individual.

According to the theory on attachment style, secure is the ideal we should strive towards (both on a personal level and otherwise). As a fearful avoidant you should avoid people with avoidant attachment styles (the most), as this exacerbates your own notions which may be detrimental to the relationship. With secure people you feel secure.

Working on yourself regardless of the person and the attachment style, I think, would mean you create some sense of security born out of yourself. If you're afraid to lose someone, you can take comfort in things like something you are passionate about, or other fulfilling relationships or goals (theoretically, as examples). So even if the person leaves you (which is the ultimate fear), you can still fall back on things. This should give you some security or at least comfort if things were to actually turn out a certain way.

I've studied attachment styles a bit, and that's what I've read and heard professionals recommend.