I am Cherelle and I am a linguist and my background is in languages myself. I speak 3 total and a partial 4th.
I completely understand where you're coming from as I'm speaking different languages with different people almost consistently. I know that my accent is not completely native in both of my other languages, but the point is, is that, I speak the language, and the native speaker of what I'm speaking is actually understanding me. This means, communication is still happening. For English, it really doesn't matter what accent you "pick" or "choose" (which that can't happen really unless you "choose" to go to the US and stay there and practice there to get a "standard US" accent). If you're talking with a person or a group of people that do not like your accent for some reason (I know with English, those do indeed exist unfortunately), forget their criticisms because there are multiple ways to speak this language. There's different accents from native speakers of it! It's the same with every language.
I'm from the Midwest region of the US but my family comes from the rural south of this country, we speak differently, but we understand and can still communicate with each other. I have a friend that has a multi-dialect accent, meaning her family who is Irish and English on one end and then Scottish and US on the other end was raised around multiple (as you see) accents and she has a mixture of different English-speaking ethnicities. That influenced her accent, but you know? I still can communicate with her and she lives life fine talking to other people.
To conclude, it doesn't matter if you can communicate. You are French yes? Sure, there's an accent when you speak English or if you speak another language but it does disappear depending on where you go and if you stay there long enough; your French accent within speaking English will lessen. That's happened to me in different countries I've lived in over time and I sounded less like an English speaker speaking these other languages.
This is a great question.
I hope this helps. Let me know if you have any other questions. I can help and work with you if you have any queries.