Well, every therapist will be different, some (I'd say especially older ones) prefer to be contacted via phone, others like me definitely prefer e-mail. Writing in English is fine, I do therapy in English, but I'd actually like to get an email written in german, as it would make things easier for me as in my experience therapy with native English speakers with no other language skills was harder, don't know why, but that's why I would like to know that you're learning German. So your email is perfectly fine. People all the time write mails with information about why they need therapy, but actually that doesn't help. You can do it, but at the end it makes no difference.
Sidenote: It might be very difficult to find a therapy. Some colleagues have waiting lists, some just say: Ask again in three months. I do have a waiting list and the duration of getting a place is about 1,5- 2 years right now. Corona did not help here. Most therapists with Kassenzulassung i know would not prefer a "Selbstzahler" over someone with GKV, but as I said, therapists are different. However, there are private practices (you can find some on therapie.de), which would have no problem with a Selbstzahler. The costs might vary, might some take the highest amount which would be 153€ in CBT, at least you would pay 100€ in CBT for 50 min. Tiefenpsychologische Therapie would be a little bit less.
There is also something which is called Kostenerstattung. For this option, which would mean that TK pays for a therapy in a private practice, you would need to get in contact with therapists with Kassenzulassung (you could search on the website of the Kassenärztliche Vereinigung of your federal state) before. And in case you do not find a therapist with kassenzulassung within maybe 3-6 months then you could ask the TK, if they know what to do or if they would pay for a therapy in a private practice. I would recommend searching the internet for an overview of this procedure. You might use "Kostenerstattung Psychotherapie Privatpraxis" as key words or start here: https://www.bptk.de/wege-zur-psychotherapie-jetzt-auch-auf-englisch-und-tuerkisch/
Hope, that helps.