Email under my Dad's name

I'm using my own apple ID for my icloud email on my phone and computer. But the recipient of my emails think my name is Martin as my emial is under that name. I changed it in settings in icloud which says my real name, not Martin, so that's not the source. It seems that my name is correct everywhere in settings, until my recipients tell me, I'm under the name Martin. My Dad's name is Martin, he gave me his old phone, but I changed the name on it. And this computer is new and has always been mine. Although there is a strange problem with the texts I send to my Dad, most of the time we don't recieve each others texts, he said I'd have to restart my whole phone to fix it but I've had it too long now and I've never been bothered to do it. Is that the issue?