I'm a native speaker of American English. No one here would say "promish" for "promise." I think what you're hearing is "promise" and "you" being elided. That is, the speaker is blending the two words together in a way that other native speakers wouldn't even notice. For example, instead of "Did you eat?" many Americans blend the words so it sounds like "D'jeet?" In the "I promise you," the speaker would have to put a break in between each word, right? And if you're speaking quickly and not being picky about enunciation, you get something like "I promishoo."
I hope that helps!