1. This morning, I walked to school, feeling embarrassed.
2. This morning, I walked to school, while I was feeling embarrassed.
3. This morning, I walked to school, while I felt embarrassed.
4. This morning, I walked to school, and I felt embarrassed.
5. This morning, I walked to school, and felt embarrassed.
6. This morning, I walked to school, and I felt embarrassed at the same time.
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#1 is the original one. What about the others? Which ones are similar to #1? Can we use the ing form about the verb 'feel'?
1 and 5 are the best, but you'd need to remove the comma in 5.
All the rest are technically correct, but are wordy and/or awkward.
All of the sentences you provided are grammatically correct and convey a similar meaning. They all express that the action of walking to school happened while the person was experiencing embarrassment.
Regarding the use of the verb "feel," all of the sentences use different forms.
In sentence #1, the present participle form "feeling" follows the verb "walked."
Sentence #2 uses the past continuous tense, with "was feeling" which suggests an ongoing action in the past.
Sentence #3 uses the simple past tense, "felt," which indicates a single past action.
Sentence #4 uses the coordinating conjunction "and" to join the two actions, "walked" and "felt."
Sentence #5 combines the actions in a similar way to sentence #4, but omits the subject pronoun "I" in the second part of the sentence, which is grammatically acceptable when the subject remains the same.
Sentence #6 uses the conjunction "and" but adds the phrase "at the same time" to emphasize the simultaneous nature of both actions.
So, in conclusion, all of the sentences are similar to #1 in meaning, but they use different forms of the verb "feel" and vary slightly in emphasis and structure.