Does registered company name matter?

I've been working on a medtech idea for a couple of years, even ran a trial with a company in Europe through my old contacts, and I have a trademark registered with the classes to cover where I'm taking the product. Now I'm ready to make it my full-time thing and I've just convinced an old colleague to join as CTO to get the product more polished and scalable for a real market launch. My CTO will need a salary in six months, and we'll definitely have some legal and regulatory costs that will exceed my resources, so my thought was to register in Delaware to prepare for future VC funding. I just realised that someone has registered a C corp in Delaware with the very same name, although the product is a SaaS and totally unrelated to what we're doing. Does it really matter as we have a trademark that covers our product in our market? Can I just add Medical or Technology or something like that to the name that we hope to trade under? Or should I just register in California (it's where I live and the name is free here) and think about incorporation in Delaware when and if it's actually asked of us? Thanks for any help in advance.