Hi Emil, how are your screencasts going? I had the EXACT same problems. I'm Brazilian, and I can't hide my accent. Additionally, I also do some incorrect pronunciations sometimes. But I tried: I created a screencast series on youtube (www.youtube.com/c/reactcasts) and it's been a success so far.
My tips are: 1 - Prepare a script. Even if you're going to change a few things while recording, having a script upfront has helped me look for correct pronunciations and rehearsal a little bit before recording.
2 - Record only voice before the video, and learn an audio editing tool such as Adobe Audition: It's always easy to sync the video later, but being able the edit just your audio recording separately will let you talk more calmly and confidently - if you say something wrong, just repeat the last sentence again and later you can cut the wrong part.
Some audio recording tips: 1 - Yes, invest on a good microphone. I doesn't need to be a professional grade one, but a good cardioid USB mic makes difference. I have a Widget B, and I like it a lot because it has good quality and built in shock mount + pop filter. Blue microphones are also excellent (both snowball and yeti).
3 - Record on a small room, and throw a few pillows around the desk and behind the mic to diffuse the audio and get better acoustics.
I hope it works for you!