US pronunciation is generally rhotic, ie. they will pronounce the r's distinctly in those words. Most of the UK (the southwest being the largest exception) is non-rhotic, as are Australia and New Zealand, and in those accents those words generally would be indistinguishable.

Certainly in the UK, 'or' and 'awe' or very similar. But I'm pretty sure the same is true across the states. The 'awe' in 'awe, shucks' is not even really a word, just a sympathetic utterance in the same sort of class as 'um' and 'aaah', so I don't think it has an official spelling, just a phonetic approximation. It is certainly distinct from the word 'awe' that's the root of 'awesome'.

Edit: I meant to point out that I have never seen 'awe, shucks' written before, always 'ah' or 'aww', but I can see how it might be written that way based on a Midwestern accent.