And what, pray tell, is wrong with that venerable complimentary close? It can even be shortened to “yr obt svt” for the young whipper-snappers of the Twitter generation.
I have the honour etc., Apollyon.
The evidence is that I’m noticing it more. 
I agree with re-arranging a sentence that just refuses to be de-awkwarded. I prefer a nice flow over stretching my brain to try to make things work that resist working.
Only the author can determine whether the sentence still says what it is supposed to say. Was it just a coincidence, or intentional? Was passive voice a conscious choice to accentuate the object of the action or a (dubious) attempt to sound formal? All of this is important.
That said, I do believe you can change the wording without changing the emphasis: “Enclosed is both the original and a copy of the Joint Defense Agreement, which Ms. Soto and I have both signed.” (The word both makes it sound better to my ears. And I followed the style advice in the rest of the thread.)
“Hey, here’s yourself’s stuff.”
Way over-complicated and incorrect. The OP’s sentence and Knead’s are the same thing. They both state that the paperwork has been signed, and the original and a copy are enclosed. “Ms. Soso and I have signed the Joint Defense Agreement” does not imply that everything’s all wrapped up and the job is done. It states unambiguously that they’ve signed the document, and nothing else. I’m not even sure how the sentence implies anything else to you. “We’ve signed this. Here it is with a copy.”
The OP’s sentence could also be read to mean that a copy, an original, and I are all in the envelope!
I have the honour to remain, my dear Dopers, your most humble and obedient servant,
Elendil’s Heir
Scarlett’s right
– Acsenray, in-house and freelance copy editor