what are they part of then?
If you can separate a thought between you and not you, then can you find a boundary? What does the boundary look like?
Remember, we are only speaking from our direct experience of the world. Thoughts are not an experience of the world. If we relax that restriction, we can say that thoughts are a kind of auditory hallucination. Like all hallucinations, they do not represent anything in the world. They are, however, represented by something in the real world, that being the electrical and chemical activity in the brain.
The separation between me and not me, is that I am my head and my body, and everything else is not me.
What does the boundary between a thought and my head look like? If by "thoughts" you mean the auditory hallucinations, that boundary is only in the definitions of the words, it is not a physical boundary that we can look at.
If you mean the chemical electrical activity, that is a part of me, so there is no boundary.
How exactly does the thought "I am my body and my head" hold up when thoughts are not you?
There's no contradiction there. If we say "My thoughts are me", then that would contradict the statement "my body and my head are me". But I didn't say that. I said I am my body and my head, and I am not my thoughts. Perhaps it's wrong to say that those hallucinations are "mine", since they are not part of me, nor part of anything.