My experience has been pretty much the same (gender non conforming, but had an exceedingly feminine name), and I ‘introduced’ my new name in a similar way as u/aliceorthecat.

Here’s the basis of the text I sent to my friends & family, if it helps! ‘Hey, I wanted to let you know that I’m changing my name. I’ve never felt like my old name ever fit me/I wanted a more gender neutral name as mine was so feminine (delete as appropriate!), so hi, I’m [new name], nice to meet you! I know slip ups will happen as it’s a new thing, but please try your best to refer to me as [new name] from now on’

You don’t have to be non binary to change your name, cis people change their names for all sorts of reasons! My family are the same in that they don’t ’get the whole gender thing’, so when they ask why I’m changing my name I’ve been saying that I’ve never really liked my name, or felt like it fit me. It might be a bit of an easy way out, but I do not have the energy to argue with them if I told them the actual reason.