I chose "balkanization" to connote the things you reference; the atomic quoting of verses of scripture does violence to the text.

Verses are definitely not functionally paragraphs. Some aren't even complete sentences!

At least in the old testament verses range from sentences to paragraphs, have any interesting examples of a non-sentence verse ?

Sure - take a look at the first chapter of Paul's first letter to the Corinthians. The sentence:

Jews demand signs and Greeks search for wisdom, but we preach Christ crucified, a stumbling block to Jews and foolishness to Gentiles, but to those who are called, both Jews and Greeks, Christ the power of God and the wisdom of God.

is divided into three (!) verses. That sort of thing is pretty common with the periodic structure of Paul's sentences. English readers, with our weak clause boundaries, struggle with those sorts of sentences anyway.

Corinthians is not part of the old testament so I'm afraid its versing might not be per the union standards.

It is currently versed as:

  Jews demand signs and Greeks search for wisdom.

  But we preach Christ crucified - A stumbling block to Jews and foolishness to Gentiles.

  But to those who are called, both Jews and Greeks, Christ the power of God and the wisdom of God.

Which seems like three complete sentences forming a paragraph to me even if not to any modern style guide.

Alternatively it could have been:

  Jews demand signs and Greeks search for wisdom - but we preach [the words of] Christ crucified  !

  A stumbling block to Jews and foolishness to Gentiles.

  But to those who are called, both Jews and Greeks, Christ the power of God and the wisdom of God.

In both cases the logical structure remains the same and the three sentences form a larger idea.

Ah - apologies - I misread you. I'm not sure whether there are any mid-sentence verse breaks in the Old Testament. I'll look!

I believe the first verse in one chapter in Job is “he said”. However, it is generally not separated out. Instead one and two are typically combined together and the next number is 3, or something like that.

There are several chapters in Job with the first verse being roughly "And X replied, saying" with the rest of the chapter being their reply. Still very much a full sentence that identifies the speaker.

I was thinking of Job 3:2, which, in some modern English translations is simply rendered “He said:”. But the Hebrew is more complete.

King James version also gets it right, but even "He said:" is a complete sentence.