I doubt you broke anything. Most higher mileage rear ends only spin one tire during a burnout because the clutches in the rear end are worn out. You could probably use a rebuild. I'd suggest getting the carbon fiber kit. It's much more aggressive and supposedly lasts longer. It's what I have in my rear end and it locks up great and spins both tires.
I don't remember off hand in which order we did them. But you can add or alternate the carbon clutches and the steel clutches in a different way to make it slip more or less depending on how you want it to react. Mine is tight and packed in there with an extra carbon clutch I remember that for sure. I wouldn't recommend it unless you're sure you want it to behave a certain way. You can google they ways to do it. Lots of information out there about it. Truck guys do it for sure.