1. I walked to school, feeling embarrassed.

2. I walked to school, and I felt embarrassed.

2-2. I walked to school, and felt embarrassed.

2-3. I walked to school and I felt embarrassed.

3. I walked to school, while I felt embarrassed.

3-2. I walked to school while I felt embarrassed. (no comma)

4. While I felt embarrassed, I walked to school.

5. While I was feeling embarrassed, I walked to school.

(Are they all the same? Would you check the punctuation and the tense?)

1, 2, 2-3, 3-2, 4, and 5 are all grammatically correct.

1 is clearly the best, but 3-2 is not far behind!
2-2 would be fine if there were no comma.

The sentences you provided have slight differences in their punctuation and tenses, which can subtly change their meaning. I will go through each sentence and explain the nuances of each one:

1. "I walked to school, feeling embarrassed."
In this sentence, the present participle phrase "feeling embarrassed" is used to describe the action of walking. The use of a comma indicates that the feeling of embarrassment is happening simultaneously with the action of walking.

2. "I walked to school, and I felt embarrassed."
This sentence uses a coordinating conjunction "and" to connect two independent clauses. The comma before "and" separates the two clauses. The first clause is "I walked to school," and the second clause is "I felt embarrassed." Both clauses are in the past tense.

2-2. "I walked to school, and felt embarrassed."
This sentence is similar to the previous one but does not have a comma before the coordinating conjunction "and." This lack of punctuation can suggest that the feeling of embarrassment is a direct result of walking to school.

2-3. "I walked to school and I felt embarrassed."
This sentence omits the comma but keeps the conjunction "and" between the two independent clauses. The meaning is similar to the previous sentences, emphasizing the connection between walking and feeling embarrassed.

3. "I walked to school, while I felt embarrassed."
This sentence uses a subordinating conjunction "while" to introduce a dependent clause that describes a simultaneous action. The comma before "while" separates the main clause "I walked to school" from the dependent clause "while I felt embarrassed." The tenses are both in the past.

3-2. "I walked to school while I felt embarrassed." (no comma)
This sentence is the same as the previous one but omits the comma. Without the comma, the feeling of embarrassment is more closely connected to the action of walking, suggesting that the two are happening at the same time.

4. "While I felt embarrassed, I walked to school."
This sentence starts with a dependent clause "While I felt embarrassed," which describes a condition or circumstance. The main clause "I walked to school" follows after the comma. The tenses are both in the past.

5. "While I was feeling embarrassed, I walked to school."
This sentence uses the past continuous tense "was feeling" to emphasize that the feeling of embarrassment was ongoing or in progress at the time of walking to school. Similar to the previous sentence, it starts with a dependent clause and is followed by a main clause separated by a comma.

While the overall meaning of these sentences is similar, the various uses of punctuation and tense give them slightly different nuances.