"With my/their/our V-ing..." as supplement to main clause

Here are some news article examples containing 'with my/their etc. being...' as supplement to a main clause:

(1) Since the opposing counsel would be the U.S. Department of Justice, and with my being new to that game, I enlisted as co-counsel an accomplished federal litigator with extensive experience in dealing with the DOJ. (Source)

(2) Since this raised many leftists' ire and with my being the reasonable man I am, I propose a compromise: no immigrants from the Third World or the Old World. In other words, no immigration, period. (Source)

(3) The Jordans were serving at Destiny Life Church in Oakland, and with their being hired by Divine Hope Church, both churches merged while keeping the Divine Hope name. (Source)

Is the use of genitive pronouns (my, their, etc.) well-formed and natural?

Or is it better to have accusative pronouns (me, them, etc.) instead?

EDIT

In a related question "with/without + pronoun (me vs. my) + gerund-participial phrase", I have shown this example from The Cambridge Grammar of the English Language (Page 461):

We set off again, the Rover going precariously slowly in very low gear up hills, with me staying on its tail in case it petered out altogether.

CGEL says, and I agree, that my cannot replace me in this sentence.

If the above examples (1)-(3) are well-formed and natural, how do you distinguish (1)-(3) from the CGEL example?

Solution 1:

1) and 2) are not acceptable to me, and I would guess that they are "errors" or "hypercorrections", although perhaps some speakers have internalized a grammar rule that permits me to be replaced with my in this context. (Compare the optional replacement of me with I or myself in certain contexts: me/I and me/myself variation might have been influenced by prescriptions or by imitations of prestigious constructions, but CaGEL says that the traditionally condemned uses of I and myself are grammatical, or at least not "hypercorrections", for some speakers.)

3) seems acceptable to me. I think it could be rephrased as "The Jordans were serving at Destiny Life Church in Oakland, and [with their hiring by Divine Hope Church], both churches merged while keeping the Divine Hope name" (replacing "being hired" with the gerundial noun "hiring").

There do seem to be many more examples of the construction that I said feels unacceptable:

• Well, with my being a pedagogue, as you know, my thoughts went back to my early years... (Bengt Tjellander)

• With my being a member of the Sharpe family, it didn't set well for me to be marrying beneath the family. (Robert “Digger” Cartwright)

I'm not sure about the acceptability of the following example:

• Probably with my being a recent immigrant, the idiot thought he could get away with it by tossing off a ridiculous explanation... (Celly Luyinduladio)

For comparison, the following two examples of "with my being a" feel quite acceptable to me (as would be expected, I think, since they aren't acting as a supplement to a main clause here):

• My response to Knole and Sissinghurst seems in retrospect to have very much to do with my being a white Southern lesbian (Toni A. H. McNaron)

• I mean that my mind has to extend through more stored memories associated with my being married as compared with the number of memories associated with my being a father. (Jon McGinnis)