I small breakout after accutane seems to be the norm. Give it time. I had a big nodule on my cheek after accutane. It scared me into starting the regimen. Since then I've had one cyst below my jaw line (where i don't apply the regimen) and the rest of my body has been clear. Don't panic yet, but the regimen is a good way to maintain clarity. I also find that I don't have to be as ritualistic about it as most people because my skin was pretty much clear starting the regimen.

Your new spots are a coincidence. There's still accutane in your system, and despite the significant improvement you were always susceptible to new breakouts.
The diet thing is a myth. The best way to prevent more acne is to continue with topical treatment. I recommend benzoyl peroxide and possibly include a retinoid. Also be careful with sunscreen, moisturizer, and makeup because certain ones could cause a breakout.