1. I walked to school, feeling embarrassed.

2. I walked to school, and I felt embarrassed.
3. I walked to school, and I was feeling embarrassed.
4. I walked to school, while I felt embarrassed.
5. I walked to school, while I was feeling embarrassed.
----------------------------------
Which ones are similar to #1?
Are they all the same and grammatical?

Yes, 2, 3, 4, and 5 all mean about the same as 1. However, 3 and 5 are wordy because of the use of the present progressive tense.

1. "I walked to school, feeling embarrassed."

2. "I walked to school, and I felt embarrassed."
3. "I walked to school, and I was feeling embarrassed."

All three sentences are similar in meaning as they all express that the speaker felt embarrassed while walking to school. However, they differ slightly in terms of their sentence structure and the use of verb tenses.

#1 uses the present participle "feeling" to express the ongoing action of feeling embarrassed while walking to school. This is a common way to express a concurrent action.

#2 uses the simple past tense for both verbs, "walked" and "felt," to indicate that both actions occurred at the same time in the past. The conjunction "and" is used to connect the two actions.

#3 uses the past continuous tense "was feeling," which suggests that the feeling of embarrassment was ongoing or in progress while walking to school. The conjunction "and" is again used to connect the two actions.

Regarding the remaining sentences:

4. "I walked to school, while I felt embarrassed."
5. "I walked to school, while I was feeling embarrassed."

Sentence #4 and #5 also convey a similar meaning, but they introduce a subtle difference compared to the first three sentences. In these sentences, the primary focus is on the act of walking to school, and the feeling of embarrassment is presented as an additional, concurrent action. The word "while" indicates the simultaneous occurrence of the actions.

In terms of grammaticality, sentences #1, #2, and #3 are all correct and commonly used. However, sentences #4 and #5 may sound slightly less natural because they place more emphasis on the act of walking rather than the feeling of embarrassment.