Which Mount Washington? Back East?
I'd keep at it in the White's while you're over there. Any of the Presidents are good for building the endurance required for larger peaks out here, The Presidential Traverse would be a good test for multi-day preparedness. The weather in the White's is more severe and unpredictable compared to out here I've found, while back-country and off trail navigational skills are more important out here IMHO. Summits are further from the road over rougher country.
Mount Baker gives the most bang for your buck for an "intro" volcano climb, good views, rugged and interesting (North Side) on the easy routes, hard routes as well, without the commitment and altitude of Rainier.