I think that the idea that your singing voice should sound exactly like your speaking voice doesn't always make sense because they're two different processes. Singing isn't normal- it's a performance. Singing tends just work better with purer vowel sounds, underemphasised sybillants etc if we're talking about intelligibility.

Another side of that is that in a 'by ear' context you'll probably be influenced to an extent by whoever you got the song from at least until you've been singing it a long time.   

Thirdly people don't necessarily have one accent or one strength of accent. My mum sounds almost english or at least fairly neutral to most ears cos it's been decades since she left Kerry and she's spent lots of time in non english-speaking environments. Stick her on the phone with her sisters though and you'll notice a (totally unforced) change.

I'd say the key is not to overdo/force it. If it feels fairly natural, it probably sounds fairly natural. If you start singing "last noite as oi lay drrrameing on pleasant days boi" you'll probably sound like a prat.

just my opinion mind.