To me, the issue is- can I identify with / get into the story being told? If I can, then the origin of the song, racial or otherwise, is secondary. Only if I feel that the dialect is too strong, too 'stagey' for me e.g. 'Freedom Come-all-Ye', 'Little Chance', do I leave the song alone.

One can also take the attitude, "The old guys down the pub never bothered much about the origins- gender, race, nationality- of the songs they sang so why should I? As long as it's a good song, a strong story I can get into, the dialect is not too 'over the top', then why shouldn't I sing it?"

As to adapting the lyrics in terms of rhyme and scan- try writing songs and you'll get the hang of it. For what it's worth (taking the lines as they are given above, totally out of context):-

'You can see that I am a fellow with a funny face
And the (my?) baby won't win any modelling race'

Having a 'funny face' is not, per se, necessarily racial.