People confuse everyday speech with what is grammatically correct. "More slowly" you say is "acceptable" but awkward? :DDD

More slowly is a grammatically correct option, as adverbs are formed in most cases from and adj. + -ly = slow >>> slowly.

Unfortunately most people form their opinion solely based on everyday speech, which can be accepted in conversation. However, many of those things are not grammatically incorrect.

In this case yes, both slower and more slowly are correct, but by no means is more slowly awkward.

Furthermore, someone mentioned near, nearly. Do not compare them as they do not have the same meaning, just like late-lately.

P.S. I have yet to hear a native speaker use few/little in the right context. I have met so many who use less for every example, even though the noun is clearly countable.

Do not search for your answers online, opinions are just opinions and what you hear everyday may not be grammatically correct. Consult the holy Bible of English grammar books "English grammar in use" by Raymond Murphy or Oxford Advanced Learner's Dictionary, the holy Bible of English-English dictionaries, or someone who has a Master's Degree in the English language and literature.