Sometimes, on certain songs, I start to sing in an accent that kind of mimics the original vocals. One song in which I have this problem is Three Days Grace's Let It Die. How can avoid the accent while still staying true to the original sound of the song?
Conscious voice training and feedback will help you immensely.
Try video recording yourself trying to sing with different accents. Really accentuate the movement of your lips,tongue and cheeks so you can see the effect. Memorise these different movements- visually at first, then the feeling.
Now try the song again, and when you feel yourself skipping into the default accent consciously use different muscles.
Additionally, think of singing phrases differently. As an example, singing Guns n' Roses Civil War in a different style is difficult if you keep in all of his trademark noises, but gets much easier if you use your own fills and phrasing.