Besides the writings of John, what is the biblical evidence Jesus Christ is the Only-begotten Son of God? Apart from John's writings, what is the Biblical evidence that Jesus Christ is the only-begotten son of God?

John makes it clear (such as in John 1:18, 3:18), but other parts of the Bible refer to many "sons of God", in both the Old Testament and the New.

Can anyone help me understand this?
The term "only begotten" is a translation of the Greek word "monogenes".  "Mono" indicates one and "genes" indicates kind.  So, the very word itself indicates a meaning of "one of a kind".  

In the Septuagint (the Greek translation of the Old Testament), the word was actually used of Isaac.  It should be noted that Isaac was not, in fact, the only son of Abraham.  He was neither the first nor the last, as Ishmael preceded him and the sons of Keturah came from Abraham's second marriage that followed the death of Sarah.  (Abraham did, indeed, "have many sons".)

Still, Isaac's "sonship" was wholly unlike all the others ("one of a kind") in that he was the son of the promise and came from a miraculous birth from a woman who was well past the age of childbearing.

As @Mike noted, followers of Jesus are called "sons and daughters of God".  However, this is quite different from the "sonship" of Jesus.  His "sonship" is wholly unlike any other, in that it is eternal.  Indeed, His birth into this world was miraculous as well.

So, Jesus is not the only "son" of God, but His "sonship" is unique, one-of-a-kind, and distinct from all others who are called sons and daughters of God.  He is literally the Monogenes--the One-of-a Kind--eternal Son of God.