1. This morning, I walked to school, feeling embarrassed.
2. While, this morning, I walked to school, I felt embarrassed.
3. While I walked to school, I felt embarrassed this morning.
4. This morning, while I walked to school, I felt embarrassed.
=====================
Which one is similar to #1? Where should I put 'this morning'?
If you mean you walked to school in the morning, then place "This morning" at the beginning of the sentence. If you mean you felt embarrassed in the morning, then place the phrase at the end of the sentence.
1 and 4 mean the same thing.
2 is not to be used.
3 doesn't mean exactly the same thing as 1.