As a native brazilian speaker, I think it should be: "Eu não vou me sentir responsável". I've never heard anyone using "me" before the verb (only in poetry). It's still correct, but sounds weird to me.
This is indeed how it is said in Portugal. Most of the time, Duolingo provides the Brazilian version - I think this is the first time I've seen the version from Portugal used. Ideally, both should be valid...
They actually use brazilian portuguese most times, portuguese from portugal has totally different accent and it's harder to understand.
Both answers are accepted in the "reverse sentence" (translation from English to Portuguese).
I think "I won't hold myself responsible" is more natural for the English, but it's not accepted. Is there a substantive difference between that and the accepted answer?
I agree with pmm123. I also felt that apart from being more formal there is a degree of deviation from the original; I guess in translation there is a matter of style and taste as well as emphasis of fidelity depending on the situation.
Its not accepted here because you are speaking in the future tense - what will happen.
I think that you need one "me", before or after the "vou", because it's a feeling = "something you let yourself feel".
If it's something external, you use sentir without pronoun.
"eu sinto sua falta" = I miss you (lit. "I feel your absence")
"eu sinto o amor de deus" = I feel god's love
"eu sinto dor nas costas" = "I feel a pain in my back"
But when it's a feeling, something more internal:
"eu me sinto sozinho" = "I feel lonely"
"eu me sinto melhor" = "I feel better"