Concerning bad/badly, the grammar police don’t agree. However, the majority seem to favor "I feel bad" over "I feel badly". Therefore, unless your sense of touch is off, you should not say or write, "I feel badly".
The reason for the opposition to "I feel badly" is because feel is a linking verb in this sense; it’s an emotion, you are not actually "feeling" anything (with your sense of touch).
Linking verbs (also called copula or copular verbs) are verbs that "link" or connect the subject (noun or pronoun) to the complement. (A complement is the part of a sentence that follows the verb and completes the sentence so that it makes sense.)
List of Common Linking Verbs:
Bad is an adjective.
Badly is an adverb.
Adverbs modify (limit, or describe) verbs, adjectives, or other adverbs . They answer the questions: "When?", "Where?", "Why?", "In what way?", "How (how much, how often)", "Under what condition", and "To what extent?"
Adjectives modify nouns and pronouns. Adjectives answer the questions: "What kind (of)?", "How many?", "Whose?", or "Which?" An adjective that follows a linking verb and describes the subject of the sentence is called predicate adjective. (Example: The moon was bright.)
Action verbs take adverbs. Linking verbs take predicate adjectives.
Tom sounded hoarse. – Sounded is a linking verb and hoarse is the predicate adjective.
The horn sounded loudly. – Sounded is an action verb, loudly is the adverb.
1. Replace "bad/badly" with "sad/sadly" to determine which is correct.
Correct – I feel bad. I feel sad.
Incorrect – I feel badly. I feel sadly.
2. To determine if a verb is linking or action, replace the verb with a "being" verb. If the sentence still makes sense, then it is a linking verb.
Example: "I am bad" makes sense but not "I am badly."
*Instead of singing, "I feel sad when you’re sad" sing, "I feel bad when you’re sad."
Incidently, the following sentences are also correct:
The food went bad. (Here went is a linking verb, bad is a predicate adjective describing food)
The battle went badly. (Here badly is an adverb describing how the battle went)
"The controversy over feel bad and feel badly has been going on for more than a century, and since its beginnings lie in two opposing prescriptive standards—that of the 1869 handbook prescribing feel badly and that of the 20th-century schoolbooks prescribing feel bad—it is unlikely to die out very soon. People will go on about as they do now—some differentiating bad and badly, some not, some avoiding badly, some not. You can see that the question is not a s simple as it is often claimed to be, and, with those considerations in mind, make your own choice. Whatever it is, you will have some worthy comrades and some worthy opponents." –Merriam-Webster’s Dictionary of American Usage
"Bad is often used as an adverb in sentences such as The house was shaken up pretty bad or We need water bad. This usage is common in informal speech but is widely regarded as unacceptable in formal writing. In an earlier survey, the sentence His tooth ached so bad he could not sleep was unacceptable to 92 percent of the Usage Panel.
· The use of badly with want was once considered incorrect but is now entirely acceptable: We wanted badly to go to the beach.
· The adverb badly is often used after verbs such as feel, as in I felt badly about the whole affair. This usage bears analogy to the use of other adverbs with feel, such as strongly in We feel strongly about this issue. Some people prefer to maintain a distinction between feel badly and feel bad, restricting the former to emotional distress and using the latter to cover physical ailments; however, this distinction is not universally observed, so feel badly should be used in a context that makes its meaning clear.
· Badly is used in some regions to mean "unwell," as in He was looking badly after the accident. Poorly is also used in this way. In an earlier survey, however, the usage was found unacceptable in formal writing by 75 percent of the Usage Panel." – http://www.ask.com/reference/dictionary/ahdict/144535/badness