There is no simple answer to this. What native speakers say--even educated native speakers--doesn't fit any logical rules of grammar. The same problem occurs in an even stronger form with "feel good" and "feel well." "I feel good" means "I am happy, joyous, optimistic, energetic." "I feel well" means "I feel healthy, I am not sick." In a formal situation, where you want to certain you are using grammar, the only solution is to express your meaning in some completely different way. This is the answer to many difficult usage questions: rewrite the sentence and sidestep the issue. For example, "I am sorry about X" instead of "I feel bad about X." The American Heritage Dictionary offers a usage note: "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. · Note that 'badly' is required following 'look' when it modifies another word or phrase in the predicate, as in 'The motorcycle looked badly in need of repair.'"

This controversy has been going on for centuries and is likely not to go away soon. The correct usage is "I feel bad." To feel badly implies that your sense of touch is not right. When you are referring to a sense of touch, then badly is used as an adverb describing the verb to feel or touch. References: https://owlcation.com/humanities/Grammar_Mishaps_bad_vs_badly https://www.merriam-webster.com/words-at-play/do-you-feel-bad-or-feel-badly

I don't know why Pimsleurs includes that! No, it's not correct, unless it is in the context of saying that someone's sense of touch is impaired. As you say, feel should be used with the adjective (bad).

thank you all for your answers!

People will often say that they feel badly about something they've done. You'd be well advised not to assume that your single authority, one who died more than 40 years ago and spoke American English, will always guide you accurately. Nevertheless, on this occassion perhaps he has. There is, as you know, no "Academy" or single defining authority over what constitutes correct English, and it's unbecoming to a non-native speaker (with several mistakes in his post) to let everyone know he thinks he knows better than most native speakers. Indeed, you might say I feel strongly about this :) On your specific topic, Merriam-Webster does not agree with the strict "you can only use bad after feel" rule you propose: "But some people make a considered distinction between feel bad and feel badly, choosing feel bad when feel is about physical health and feel badly when feel is about an emotional state. Others switch them with just as much intention. These uses are established enough that some dictionaries (including Merriam-Webster Unabridged) cover badly as an adjective; it is, after all, following a linking verb." https://www.merriam-webster.com/words-at-play/do-you-feel-bad-or-feel-badly https://separatedbyacommonlanguage.blogspot.com/2006/08/badly-and-poorly.html https://www.dictionary.com/browse/badly https://english.stackexchange.com/questions/265651/i-badly-need-to-and-i-want-to-so-bad