Here is what the cabinets look like from front view. When the delivery men said they were not going to push frige in, making it easier to hook up water line, I was so foolish I didn't realize the doors wouldn't open if pushed back.

Sorry, but I don't think there are any "regular" refrigerators that will sit completely flush in your space (and still have doors that open fully). Counter-depth refrigerators (other than truly built-in models) are not really counter-depth, they're just more counter-depth than full-depth fridges. If you push it back, the doors won't open, as you noticed. You don't have the height for a truly built-in fridge unless you remove the cabinet above.

You've got the best choice for the space. All doors of fridges will have to stick out unless you want to spend 3x your budget and go for a fully integrated model. Even "built in" fridges have doors proud of the cabinetry.

counter depth refrigerators have the doors stick out- but at least you see the stainless of the doors, not the body color of the fridge itself. check out the electrolux icon counter depth. it has the option of a trim kit which goes stainless around and louvers on top giving it the look of a built in. sort of looks like the sub zero built in (not the integrated).

The Liebherr freestanding fridges look pretty darn integrated. They are worth looking at if you really want the look/function. A little pricier, but not as much as truly integrated/built-in fridges. They are narrow and are actually counter depth including the doors- and the hinges are very tight to allow for door swing.