Many trans people call the name they were given at birth their "deadnames." It's a pretty new term. As someone who transitioned later in life, and has somewhat of a public profile, I simply call that name my "old" name - it follows me a lot, and is far from dead.
But even so, I used to cringe and feel bad when I saw it or heard it. And when I investigated that feeling more deeply, there are a couple of things I noticed. First, it brings up self-doubt. Am I really who I say I am? And it brings up fear: do other people see me as I am? What if they don't?
Your own relationship to your old name or deadname is something worth diving into, because it's likely that it will arise from time to time. Old friends you haven't seen in many years may not have learned of your new name. Family members forget. Even if you've gone through all of the hoops to legally change your name, official letters or emails from one source or another will find you.
As I connected with those feelings with self-compassion, and as I have felt more at ease with myself, hearing or seeing my old name doesn't bring up the same feelings as strongly anymore. I was on a Zoom party for an old friend, and there was someone on that call who hadn't seen me in many years. She asked over chat if she used to know me as "[my old name]", and it was surprising easy to just say "Yes! Hi, good to see you!"