"I was given the picture, drawn by Jane."
Can the sentence above be accepted like this?
"I was given the picture, and the picture was drawn by Jane."
The first sentence is fine. Variations of it would be "I was given the picture, which was drawn by Jane", or "I was given the picture that was drawn by Jane", or "I was given the picture Jane drew." The "drawn by Jane" part is secondary supplemental historical information that describes the picture.
When you use the "and" construction of the second example, it elevates the importance of the second part so that you have two, equally important, standalone statements. Jane having drawn the picture is at least as important as your having been given it.
The next step in this evolution would be, "The picture given to me was drawn by Jane", or "Jane drew the picture given to me." In that case, what is most important is that Jane drew it. The fact that it was given to you is now the supplementary information.
So they're both grammatically acceptable and mean pretty much the same thing, but the nuances are different.