Native French speaker&English accent in French, why?

Hello,

I'm a native French speaker and I've been learning English for quite a few years now.
I've never been to an English-speaking country but I try to listen to as much English as possible.
I'd like to become an interpreter in a few years and that's my spoken French has to be in pristine condition.

I posted an audio file on a languages board to get comments (I didn't say anything about me, so the answers just can't be biased) and 1 out of 2 people said I sounded English in French. They weren't sure whether I had an American or British accent but they were categoric and said I just couldn't be a native French speaker.
Apparently, I butcher my rs and put the stress on the second syllable and other things too, but I listened to my recording and I just can't put the finger on the mistakes I made. I just don't hear them.
I don't sound native in English either, although people don't say I sound French either in English (the last comments I had were "you sound a bit Swedish - I never learned Swedish, go figure.)

What would you do? I mean, several people can't lie.
How is this possible, and do you think I could lose such an accent?

Thank you.

Because of background

Thanks for the answer, but that doesn't explain how I could get rid of that accent.

I can't hear anything so I won't comment

The English do tend to be a bit on the boring side, but I can't blame them with their atrocious weather and what not. I wasn't particularly interested in this rubbish before but now I can see the funny side of it. Just don't bring up any quotes that insinuates that Spanish is vulgar French which is what I expect an ignoramus to believe. The Queen's English is essentially Modern Anglo-Saxon as passed on by generation after generation of stiff necked Norman nobles with their noses in the air. English is essentially Low German plus even lower French minus any sense of culture.

Jessica, how about posting your audio file here so that we can point out specific features of your accent that you could get rid of?
As a fellow native French speaker, je pourrais sans doute t'être utile dans cette démarche.