You are definitely not alone when it comes to anxiety. I made my own thread years back xD
So if you want an encouraging story: Hi, hello, I was very anxious when booking my first lesson, and while there were some technical issues, the tutor was super patient, and I’ve been using her ever since. 
Thanks to the tutors I’ve used, I managed to pass the N4, N3, and N2, and my conversation and listening level have skyrocketed so much more and faster than I could’ve ever done by myself. Super super thankful I took the plunge.
I think you’re on the right track of finding someone who you click with. I found I was more comfortable aiming for someone who seemed to have a lot of teaching experience on the platform, good reviews, and generally knowledgeable of Japanese (with some English to help me out). (Also kind and patient! Definitely watch the videos + read the descriptions.) I think I aimed for a professional teacher at first for those reasons. But don’t underestimate the power of community tutors. I definitely found ones I LOVE, who may not be ideal in teaching the nitty gritty, but are great for other factors.
I also found that as my level grows, I realize that having a common language isn’t as important to me, to the point where I booked my first Russian lesson (happening today, oh boy!), where I requested full immersion…and I do not know Russian, haha. But that’s my adventure, I know it’s not for everyone. But mentioning it so you know to feel open to your changing tastes and preferences.
I’ve never found price to be a factor in teaching quality. It mostly seems to be professional vs community tutor. Professional tutors, from my experience, are usually more knowledgeable of the language, all the fancy grammar terms like direct object, syntax, passive, causative, etc. and can explain the differences in similar grammar/vocab more easily. They’re usually more expensive for this reason.
For justifying a private tutor: spending money is an amazing motivator, haha. I KNOW I’ll do the homework they give, I KNOW I’ll be studying in between and practicing a lot, because I’m not only paying my hard-earned money for it, but also because I’m spending their time. And the one-on-one aspect means I get 100% of the attention and specific help I need. If I want to group study, I can always do that on the side. And like I mentioned, the speed at which I was learning was nothing like what I was getting before I started using tutors. I know everyone is different, but it was really life-changing for me.
Good luck, you got this! <3