I'm glad you were able to choose a name! Clay is a really nice name.
I just wanted to add something here that I did when choosing my name.
What I originally did was write down a list of names I liked and go around, asking people to place a check mark beside the ones they like best. Once I dwindled those down and made sure I liked them, I made another list and did the same.
However, I didn't end up going with any of them. I chose Maksim (mostly used as Max) because it was very similar to my birth name (Madison) plus it helps me stay in touch with my Russian roots.
As far as middle names go, originally it was going to be Ashton (derived from Ashley), but I decided that I really didn't like that, so I asked my parents what I would've been named if I was born a boy, and they told me it was Nikolas, so I decided to go with that as my middle name! So from Madison Ashley, I went to Maksim Nikolas (though again, most people call me Max).
Basically, it can be anything you want it to be, and the process can change from person to person. Like your mom, my parents were both very supportive of me throughout this entire thing so I wanted to help make it easier for them.
However a lot of people do prefer to make a complete change with their name, breaking away from everything, even initials and sounds. It's really up to you, but it sounds like you've decided, so that's great!
Welcome, Clay.