There is a difference, as Jim says, between accent and dialect. Dialect is more problematical, as it entails using different vocabulary, verb form, etc, and the rhymes and scansion may depend on it. And dialect won't work without the right accent, it will sound ludicrous.

Many Scots songs, though, are in something close to standard English, and don't require much tweaking to be singable in an English, Irish or American accent if that's what comes naturally to you. Sometimes they are written down in a phoneticised approximation of a Scots accent, which makes them look more Scottish than they sound. There are other songs that exist in a Scots and a standard English version (e.g. Braw sailin' on the sea/ Fine sailing on the sea); there's nothing to stop you making an adaptation like this if the end product sounds consistent and stands alone.

So choose your songs carefully and think how they would sound in youur natural accent. The occasional dialect word may be necessary, and can be left if required, but if the rest of the song can be sung in English, then that's going to sound far better than phony Scots.

Look at it in reverse: if Scots or American singer decides to sing "Down by the Banks of the Sweet Primroses", you'd think it odd if they either attempted a southern English accent, or changed "fair maid" to "bonny lass".

Marje