Focus on shade plants that like moisture and forget the raised bed.

There are gazillion native plants that like moist soil and partial sun. Just add a bunch of compost when you dig.

Since you have projects going on in both the front yard and backyard also, perhaps you could use the lasagna method to improve the soil in this side yard and plant in the fall or next spring.

Not sure what the "lasagna method" is.

There's no reason not to have a raised bed, but, it may not be necessary. I have an area in my side yard that gets muddy every time it rains. Gutters drain into it, there are a lot of trees and the land is pretty flat. Grass hardly grows because of the tree roots. I plant water loving plants like hydrangea and clematis.

Visited my house today (not moved in yet, remodeling not quite done), and that side yard was REALLY muddy after a rain/snow episode yesterday. I don't think even clematis or hydrangea would like such wet feet. I really think I'll have to do a raised bed. If it got more sun there, I'd love to do a veggie garden, but as I stated, it only gets morning sun. I'm thinking my hostas and heucheras would like this less-sun location, along with my painted fern.