Holding my hands up, and possibly causing an outcry - I don't know this tune so cannot say for sure.... but...

It sounds to me like you're hybrid picking - using pick and fingers. Chicken pickin' is the use of thumb and fingers (it's called chicken pickin' because the hand position kinda looks like a chicken's head). Gives you what I'd call a hillbilly kind of sound... I know Zakk Wylde does it, but he's a bit of a hillbilly!

At a guess, is the fret buzz being caused by your frettng hand fingers not being close to the frets as you play as you're concentrating on the other hand?

There is nothing I can think of with hybrid or chicken pickin' that would cause fretbuzz... to fix it you're gonna have to go slow and figure it out. I can say for sure there are no secrets with picking technique, it's just practice to refine your skills....