My experiences, for what they're worth...
(1) I'll dig in with my snowshoes, to go up a hardpacked, steep ramp, or if there is sufficient trail width, try to zig-zag. Ditto the way down, except that I alternate between pushing down on my toes and kicking back on my heels to control my momentum, depending upon snow consistency, etc. I don't slide backwards on the uphill sections because my Atlas 1200 series snowshoes have lateral teeth in addition to the bear claws, and this tends to keep my planted foot in place as I move the opposite foot forward.
(2) Typically I switch to crampons, unless it's a very small area of ice flow.
I don't trust my snowshoes to pin me entirely, or more to the point, I find it more taxing to do it with snowshoes than crampons. I tend to walk duck-footed, toes splayed outward, and lean on the flat sections with as many points as possible for stability, and with front points on more vertical stuff, or zig-zag as before if the trail and ice conditions allow.
Frankly the hardest part for me is wearing snowshoes or crampons towards the end of the day, when I'm really tired, walking down off a trail, when one needs either of the forementioned, and stablicers or micro-spikes won't do. With snowshoes, I find myself skiing and on rare occasion, slightly out of control, because I took my attention off the task at hand for a nano-second. With crampons, the issue is to avoid front pointing one's self, or my personal favorite: snagging a gaiter or crampon strap on the opposite foot and a one point face plant for my fellow hikers to chortle about later on.
On a recent December warm-up trip down the relatively easy trail off Mt Tom/Field I misjudged a slope during a zig-zag with snowshoes. Not a split second later I was flailing on my side/back and then self-arresting with my hiking poles to avoid sliding all the way down a slope off the trail. One could argue that it might have been wiser to ditch the snowshoes on a fairly hardpacked trail at this point... ...but there was group think that snowshoes were needed on the whole trip.
Good luck with whatever you try...
--Mike