To me, "soul" (personal expressiveness) is the essential ingredient.

I don't think it's necessary to be so very "natural" that you can't use a different approach for songs from different traditions. I think it's entirely acceptable to sing songs with different origins ~ say, hillbilly songs, blues songs, and Irish ballads ~ and to utilize a different style and even a different "accent" for each. What's important is to convey genuine feeling in each context without being primarily concerned with play-acting, that is, without trying to be someone other than yourself.

The human self can be pretty complex, and you shouldn't feel overly restricted to maintaining your ordinary workaday indentity as the single overly-obvious feature of your singing voice. I feel quite sure that my blues-singer self is an entirely valid aspect of my real self, just as much as my country-voice personna and even my not-entirely-authentic "stage-Oirish"-accented balladeer self. They're all perfectly valid "flavors" of a single complex musical personality.

Now, I'm not saying that all of these "voices" of mine are of the best quality, and certainly not that they're equally good. But they're valid as "folk voices": sincere and actually quite unaffected. I regard the different "accents" ~ actually, different sets of pronunciations and of vowel and consonant sounds ~ as aspects of the various folk subgenres, intrinsic to the songs themselves.

I would also argue that various aspects of vocal quality (head vs chest, nasal-ness,. etc.) are features of different folk traditions in exactly the same way as different approaches to English pronunciation (i.e., different accents).

I will use different vocal styles to sing Mississippi John Hurt songs, Gershwin tunes, Royal Navy sea shanties, and Hank Williams classics. But I try not to let those incidental differences stand in the way of expressing my feelings about each song and its subject matter.

I suppose the first step is to develop a way to achieve vocal self-expression however you can, within whatever style or tradition you feel most comfortable. Once you feel coinfident that you have a voice of your own, then (I think) you can branch out and bring your manner of interpretation to different kinds of songs of different styles, and all of your performances will retain an element of natural self-expression.