Will changing my URL mess up my SEO?

Hey y'all. Long story short, I changed the name of my podcast. I purchased the new domain and currently have it set up to redirect to my existing website. So basically you can get to my site with both the old and new domain names.

However, when you're at my site the URL is still the old domain name.

If I did a reversal (change the URL but keep the old domain name so it still directs to the site) would this screw up my SEO?

Forgive me if I'm not wording this correctly, as I'm not very advanced when it comes to this stuff.