Support with brand identity and higher search engine ranking

Hello. I have a website called "acupuncture at your door" www.acupunctureatyourdoor.co.uk as that is what I set it up as 9 years ago. My face book page is listed under my name, and my insta page is Tara_acupuncture. I know they all need to be called the same thing. The problem is I do a lot of different things, and I am worried that if I change the website name then it will affect my search engine results. I am a sole trader so maybe it should all be listed under my name which is quite unique with no other people with the same name coming up on google. Can you help me sort it out? So that it doesn't affect traffic?And i would like to boost my search engine rankings so that I come listed higher up for the services that I offer locally. I am an acupuncturist, massage therapist, doula and placenta specialist. I currently work in one clinic in East London but will shortly be starting in another. I also do mobile home visits which is how a lot of people find me as its unique. Any other feedback on my site would also be greatly appreciated. Many thanks