You can sound like a native.
Non-native speakers often give themselves away with their vowels, as English tends to pronounce them as diphthongs. American English diphthongizes them in a unique way. (In fact, you can tell a lot about a variety of English by its diphthongs.)
In this video I explain the Standard American English diphthongs of /ey/, /ow/, and /uw/ of my native dialect, having grown up in a middle-class family in Nebraska and Colorado.
Even though speaking with a foreign accent seems like a normal state, you can learn how to make the sounds that sound easy in the mouths of natives. This video series increases your awareness of all the parts of your mouth you use for speaking. A language never felt so good!
Dr. Thomas Coates blew my mind. He taught me how my tongue, lips, jaw, and teeth create language. Like a Chinese calligrapher learns how each finger holds a brush, like a yogi breathes with specific depth and stretch of her diaphragm, I took the first steps towards mastering language: losing my accent.