There is a difficult problem here to solve, BUT, there is absolutely nothing wrong in changing the words/dialect/pronunciation of a song into one's natural brogue. Apart from which the vast majority of traditional songs in English occur in multiple versions in almost every part of the English-speaking world so another answer is to get off one's bum and seek out a local version. In a free country like ours we are all at liberty to do what the hell we like with the songs. The audience response should be the yardstick to how successful we are.

And another way of looking at it. I sing some songs from my own area that are strongly laced with dialect, quite unlike the way I speak. The way I look at this is I am helping to preserve/promote a dying way of speech which is disappearing under the dominant city dialects and the rapid spread of SE influenced by mass media.