To expand a bit on what Yooper said, for clarity's sake, it's the matter of scale.
When you put fresh root beer and yeast in the 2L (or other) bottles, and then refrigerate when the bottle is firm, only a small amount of sugar is converted to alcohol and a comparatively large amount of CO2. Halting with cold (or pasteurization for room temp storage) will keep your beverage at the miniscule alcohol content indefinitely.
When you put it in the carboy, you were giving the yeast free rein to go nuts on the sugar, converting everything possible to alcohol, and saving NONE of the CO2, which means when you want to carb it, you'll have to add more sugar for CO2, and still wind up with a VERY alcoholic, dry root beer. If you want to backsweeten, you can, but you'll still need to either pasteurize or chill to halt the yeast on that, or suffer bottle bombs (plastic will survive better, but not forever), or use non-fermentable sugars like lactose... which doesn't seem like a great choice for root beer.
Unless your goal is hard root beer, it should never go in a carboy/fermentation vessel. Since you made it the other day, it's likely finished all initial fermentation, so please don't give it to your kids in any real volume...
I'd stick the carboy in the closet and let it do it's thing for a few weeks and see how it turns out for the adults. Meanwhile, start a new batch of root beer for the kiddies and put it in bottles, to head into the fridge as soon as they feel solid.
