I tried to "carve" on some shallower parts of blue and green runs and think I was able to sort of do it (assuming carving means going from one edge to another without fully turning). On steeper parts of blues, I have to basically traverse the whole run diagonally and then turn the other way---my linked turns are super ugly but I can see how to progress from here. Basically I think comfort with speed will be key, which can only happen with more practice.
You definitely turn while you carve, the only difference being that while carving your edge is fully engaged in the snow and the back of your board isn't dragging/slipping/speed checking. The track you leave while be a thin line of your board edge instead of the wider, messier track of a skidded turn. You're able to go a lot faster while carving and you lean into your turn/carve a lot harder than for skidded turns.
While learning your linked turns I'd suggest learning on greens and shallower blues. It's a lot better to get the technique down right than to "be able to make it down harder hills without falling." If you're not able to link turns on a hill because of its steepness/difficulty, avoid that run if possible until you're more comfortable.