I would also answer anyone who tells me to "sing in your own voice": "Well, what do you think is my own voice?"
Like you, Will, I have moved around: Scottish mother (Aberdonian but brought up in Greenock, so no idea what HER accent was), English father (Londoner but pretty "neutral" accent). I was born in Glasgow, but have lived in Essex, Suffolk, London, Sussex, Devon, Cheshire, Shropshire, Suffolk (again), and for the last 30+ years, in Scotland.
I like to sing all sorts of songs, including Scots ones, and would not dream of trying to Anglicise Scots lyrics, but rather "dae ma best" at pronouncing them a bit better than some of the cringe-making efforts you do hear at times: my mother used to correct me, but she is now long gone, so I'd rely on friends to tell me straight. My normal speaking accent is perhaps to define: English, yes, but where from? Still probably a trace of Suffolk in it, tho' I can switch it back on for songs that come from Suffolk. (While my Scottish friends think it's Devon......!)