Hi Charlotte,
It would be extremely unlikely for a Rough-legged Hawk to be anywhere in WA at this time of year. In addition, they seldom venture into Seattle even during the winter in my experience. They are strictly a northern bird in summer and are at their Arctic breeding sites this time of the year. In Alaska (e.g. the Seward Peninsula), the young hawks are fledging about now.
If your bird was a buteo, it was most likely a young Red-tailed Hawk from your description. They are fairly common in our area. Most have fledged by now in western WA and many will still be food-begging from the parents. The young have an extremely distinctive call that can be heard from a remarkably long distance.
Hope this helps...

Bud Anderson
Falcon Research Group
Box 248
Bow, WA 98232
(360) 757-1911
bud at frg.org