Saying “I feel positively about...” to praise something sounds more natural than “feel positive” to me. It’s strange that we usually say “we feel...” with an adjective but here we’re using an adverb. Can someone explain this please?
Modern Grammar considers 'to feel' as a linking verb here.
A linking verb is a verb which links the subject of a clause and a complement.
'Be', 'seem', 'become' are linking verbs as well.
Oxford Living Dictionary https://en.oxforddictionaries.com/definition/complement
defines COMPLEMENT as
"One or more words, phrases, or clauses governed by a verb (or by a nominalization or a predicative adjective) that complete the meaning of the predicate".
So, the only grammatically correct sentence is 'I feel positive about...'.
As about 'positively', it's an adverb and can be used with action or state verbs.
See some examples (https://context.reverso.net):
Their interrelation may be explained positively or negatively.
Emphasis on the health programme had positively impacted other areas.
There is evidence that investment in energy infrastructure positively influences growth.