A complement or an adjunct test

I am asked if a PP is a complement or an adjunct. I am not a native speaker. Would anyone help me to check if the following sentences are acceptable to native speakers?
The original sentence is
He drew pictures of his bag on a piece of paper with lines.
To my understanding, technically this sentence has two meanings. Am I right? We are asked if "with lines" is a complement or an adjunct.
To test if "with lines" is the complement or adjunct of the VP,
I insert "carefully" before "with lines". Then if the following sentence acceptable to native speakers?
"He drew pictures of his bag on a piece of paper carefully with lines."
To test if "with lines" is the complement or adjunct of the NP, I insert "partially". Then if the following sentence acceptable to native speakers?
""He drew pictures of his bag on a piece of paper partially with lines."
Thank you very much if you'd like to help out.

He drew pictures of his bag on a piece of paper with lines.

To me, this simply means that he used lined paper (a paper pre-printed with lines - which is not good for drawing, by the way!)

So you could say, "on a piece of lined paper."

However, if you want to say his drawing was composed of lines, then place "with lines" near the word "drew" to indicate "how" he drew.
He drew pictures of his bag with lines.
"He drew with lines pictures of his bag" is awkward.

But this could also mean the bag was decorated with lines.
He drew pictures of his bag with lines.