I am an American and I definitely do not always use the AYE pronunciation, and in the example you provided I would use UH in place of each AYE. I suspect that about 99% of American-born speakers would do the same.

Wasn't Jimmy Shtewart transforming 's's to 'sh's many decades ago on the silver screen? I seem to recall Dubya also doing it, and in the comic strip Mutts, nearly every 's' is pronounced 'sh'.