Chocorua is an excellent way to get some winter climbing experience. I've done the Piper Trail in winter and it is a great hike. You are below treeline and protected from the west until the last few hundred yards. If you're there in the wind it will give you some idea of what being on Washington can be like, and you can get down out of the wind easily whereas on Washington you can't. The craggy summit cone is usually scoured of ice and snow, but there may be patches you have to cross to summit, so you may or may not get a chance to use crampons.
If considering Carrigain keep in mind you will have to hike in on Sawyer Road adding considerable distance and time to the hike. I still think if you are interested in getting some winter, above-treeline experience the Southern Presidentials are the best place to learn. The Pierce-Eisenhower Loop will give you a feel for most things you will encounter in a less harsh environment than Washington.