'Me sitting' can be regarded as having 'me' as the object of 'mind' and 'sitting here' as a modifier of me; 'my sitting here' as a phrase can be regarded as the object of mind. Both in practice mean the same. The same applies to us/our in these circumstances, so it's not just a mispronunciation.
"Oh, my!" is almost certainly shorthand for "Oh, my God!" (or whatever abstract concept to which you wish to appeal under stress.)
All of this is incredibly simple and can be demonstrated briefly by example, with no obfuscation or explanatory sites needed.
"Me" is an object pronoun, both direct and indirect. "My" is a possessive pronoun.
1. "He saw me." ("Me" as direct object pronoun.)
2. "He gave the book to me." ("Me" as indirect object pronoun.
3. "That's my book." ("My" as possessive pronoun.)
Yes, "MY sitting here" is correct, simply because one owns one's decision to plunk one's body down somewhere. A simple, straightforward use of the possessive.
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