I'm doing it lifting my foot. The crosscuts is the more advanced version. Once skaters get used to doing it the "conventional" way, they'll gradually move on to do crosscuts i.e. no lifting of foot, just rapid "scissors" motion. I'll like to do the basic first, instead of trying the advanced way.
Interestingly for me, I found (and still find) the crosscuts easier than the basic crossovers.
I learned the crosscuts first, because basic crosscuts don't require as much edge control as basic crossovers, since you have two feet on the ice most of the time. My coach told me back then: "try to make a number 8 on the ice". 2 years later, I'm still working on it, but more on power and form.
For the basic ones, what really helped me was practicing backward outside edges. The weird thing is, in the ISI system, backward edges are more advanced than backward crossovers (and yes, they are harder, in my opinion).
If you're stopping while doing the crossovers, most likely your weight is nearer the toepick. Keeping the weight more in the middle (for me, I
intend to put it on the heel, although it really does end up in the middle) helps in controlling the edge.