Your pronunciation can most certainly become native. The easiest path there is to immediately abandon all attempts to speak the language as if it is your own, and focus on the rhythm and intonation of the language. Also, pay attention to the shape of your mouth when speaking. I almost feel as if I have an entirely different voice when I speak French than I do in English. Focus on the sound and rhythm of the language as much as you focus on the vocabulary and grammar.

When I was first learning French 22 years ago, my teacher, an older woman who'd grown up and studied in France, shared a funny little joke about French women: their wrinkles start from the lips and spread outward across their face. This was said because they keep their lips far tighter, nearly pursed, than one does when speaking English. I've noticed and paid attention to how the French use their mouths when speaking ever since. It's been immensely helpful.