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!

I grew up minutes away from the Canadian - United States border. It was a 15 minute drive for me to make it to Niagara Falls. I have to parrot Space Cat's response. For me, in my over 70 years of interacting with Canadians and Americans I have not heard this 'ish' as in promish. Unless it was intentionally done on purpose by movie actors. I don't know? Do we sound like that to folks overseas? 🤷‍♂️

Hmm interesting but I've never heard this before

I agree with Chris. I've never heard this, except possibly from drunks or people with missing teeth. Maybe the final SS sound is influenced by the sound that comes after it.