Sherman wrote with great gravity because the occasion was one of great gravity. If you are explaining to someone why you are going to order your army to burn their city down, I think that excludes what you are going to say (or write) from the realm of "everyday speech."

Of course not all speech is everyday. Here is Stephen Douglas, in the second of the Lincoln-Douglas debates, laying out what would later be called the Freeport Doctrine (basically, leaving the slavery question up to the individual states):

"It matters not what way the Supreme Court may hereafter decide as to the abstract question whether slavery may or may not go into a Territory under the Constitution, the people have the lawful means to introduce it or exclude it as they please, for the reason that slavery cannot exist a day or an hour anywhere, unless it is supported by local police regulations."

This was a debate, so Douglas was not speaking from a prepared text (although he was, as he states in the debate, expressing a position he had expressed before, in other contexts). The phrasing is (to me) unnatural and archaic in places ("It matters not" "hereafter decide as to"), and definitely shows rhetorical flair ("cannot exist a day or an hour").

Of course, a political debate is something one can prepare for, and Douglas had spoken on the topic before, so it's hard to separate out how much of the quote is "canned" and how much is genuinely off the cuff (I think you're asking about the latter).

For everyday dialogue, court transcripts are probably the closest we'll be able to get. Here is part of the transcript of the Lincoln Assassination trial. Dr. Samuel Mudd's lawyer is questioning one of Mudd's associates:

Q. You say you were not separated from him the next morning for over five or ten minutes at a time?"

A. I think not. Our business after the purchase of this stove was that we had some little things to purchase, and we separated; but I saw him frequently, sometimes opposite on the street. Once, I think, he came from the Bank of Washington: He had a little business at the bank.

The dialogue is still identifiably antique ("this stove" instead of "the stove," "opposite on the street" instead of "on the opposite side of the street"), but if someone talked to me this way today I probably wouldn't think they were putting on airs.

Just a few lines later, however, a Judge advocate objects to the line of questioning, and Mr. Ewing, Mudd's lawyer, defends it:

Mr. Ewing: May it please the Court, it is of a very great deal of consequence. The prosecution has attempted to prove by one witness a meeting between Booth and Dr. Mudd, and an introduction of Booth to Surratt by Dr. Mudd here in Washington. We expect to be able to show to the Court conclusively, that, if there was any such meeting, it must have been at this visit to the city of Dr. Mudd, as to which we are now inquiring. In that view, it is of very great consequence to the accused to be able to show that he came here on business unconnected with Booth, for the purpose of rebutting the presumption or inference unfavorable to him which might be drawn from the fact of his having met Booth there.

"For the purpose of rebutting the presumption or inference unfavorable to him" is quite a mouthful, and is definitely part of an off the cuff remark, albeit one presumably informed by years in court, dealing with legalese. All the same, though, you might find a modern lawyer throw out a similarly "unnatural" phrase, if not one with the same construction. In fact, I would say that everything in that quote up to "in that view" would not be out of place in a modern courtroom, but would be out of place in everyday conversation. So again we come to the question of occasion.

So I would say that if people in the past thought the occasion called for eloquence, and they had sufficient education to achieve eloquence, they would speak eloquently. This eloquence might seem extra eloquent to us because of the blessing of antiquity, and because we associate archaic diction with the eloquent documents and speeches that have survived due to their importance and eloquence, rather than people talking about buying stoves and seeing their friends on the other side of the street.