Dale is spot on on this one. I'll just add a couple of things.
If you want a little more room to wander without postholing around, try to swing a trip to the olympic coast. It does get a little pricey with the ferry.
Suiattle River road is a great option for a night or two, and has some great use-to-be-car campgrounds, but I would say is limited as far as scenery and leg-stretching.
The Chelan Lakeshore Trail is great in March, but again, the ferry ride ($45?) is kinda pricey. Excellent views and plenty of side trips out of Stehekin, although the bakery won't open til June!
East Bank of Baker Lake might be just the ticket. Very close (<2hrs from bham), good campgrounds, a view or two if the weather cooperates.
Thunder Creek off of hwy20 from Colonial Creek Campground is a nice wandering in March, and will ensure some solitude (and probably a hint of snow).
If you want real snow camping and the avalanche forecast is stellar, consider Herman Saddle near the Mt. Baker ski area. Short hike, great views, and it isn't Artist Point.
For quick overnights near Bham we go up to Lizard Lake or Pine Lake in the Chuckanuts. Certainly not "out there", but if it is windy you can't even hear I5! (haha)
Hopefully this gets the wheels turning. It's hard to get the hiker blood going when you can feel spring coming on but everything is still buried.