Finding the right house at an affordable price is no small feat. That might steer your decision.
As for neighborhoods, if you go to the neighborhoods to the north of downtown, you have the Burke Gilman trail which is flat and together with the trail on the east side, forms a very nice 26 miles or so of trails. Though it's paved, most sections do have a foot or two wide dirt/gravel running path next to it. Those neighborhoods would be Greenlake, which also has a nice 3 mile loop trail around the lake, Ravenna. Wedgewood, Laurelhurst, and on and on....
Really, your basically nuts if you do anything except come out and check it out for a couple of weeks, and decide. It's a big decision, and being a few miles closer to your fav trail is just one factor. And i can't imagine it could possibly be one of the most important factors, for that matter.
Welcome, and check out one of the many running clubs, too.