In the present day, I felt gloomily would be taken for an error in standard English— one says I felt gloomy. We regard feel (and various senses of other verbs of states of being like sound, taste, or appear) as a linking verb (copula) when there is an adjective complement, as the subject and its complement are being related or equated: I feel happy ≈ I am happy.
Mary Shelley's Frankenstein was published in 1818, however, when some sensibilities about the language were different. A search in the Corpus of Historical American English turns up a similar constructions into the late 19th-century:
Sarah, you feel poorly to-day. (Elizabeth Stoddard, Two Men: A Novel, 1865)
Don't call him my friend, even in a joke; it makes me feel awfully. (Edward Everett Hale, The Brick Moon and Other Stories, 1872)
Mrs. Church appeared to feel badly. (New York Times, 1884)
Sir Anthony Eden himself, in a sentence no doubt contorted to avoid ending with a preposition, once said
If there is one country about whose radio campaign and criticisms of Greece I feel badly, it is Bulgaria. (Anthony Eden, August 1945)
The Merriam-Webster blog covers the phenomenon in its post regarding the ever-contentious I feel badly:
[S]ome 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….
There was a time, mostly in the 19th century and mostly because bad was thought to only mean "somewhat evil," that people were advised to use feel badly for both physical health and emotional states. Modern advisers recommend feel bad in both contexts.