Here are Google Ngrams on the frequency of "recently" and "lately." In American English, "recently" surpassed "lately" about 1860. https://books.google.com/ngrams/graph?content=recently%2Clately&year_start=1800&year_end=2000&corpus=17&smoothing=3&share=&direct_url=t1%3B%2Crecently%3B%2Cc0%3B.t1%3B%2Clately%3B%2Cc0 In British English, "recently" surpassed "lately" about 1890. https://books.google.com/ngrams/graph?content=recently%2Clately&year_start=1800&year_end=2000&corpus=18&smoothing=3&share=&direct_url=t1%3B%2Crecently%3B%2Cc0%3B.t1%3B%2Clately%3B%2Cc0

They are interchangeable with no difference in meaning. Recently is more common. The Collins online dictionary gives "recently" as among the 4000-most-common words and increasing in frequency over the last 100 years. https://www.collinsdictionary.com/dictionary/english/recently The Collins online dictionary gives "lately" as among the 10,000-most-common words and decreasing in frequency over the last 100 years. https://www.collinsdictionary.com/dictionary/english/lately

You should say " I haven't been feeling well lately" , because lately has a continuous meaning