First check current road conditions on the Gifford Pinchot national forest website.
WA141 from White Salmon (on the Columbia) to Trout Lake is the only year around highway to this area. That's also home the Mt Adams Ranger district station.
NF23 is the direct route from US12 to Takhlakh Lake on the NW side of Adams. I don't know if that is open yet. The connection from the to Trout Lake area is partly gravel and might not be open.
WA503 from Woodland to NF90 gives access to the Lewis River campgrounds, and once snow is gone connects to 23 and Trout Lake. Wind River Rd from Carson is also paved to 90.
NF25 from 90 to US12 (Randal) is iffy. It's paved, but high enough (near Mt St Helens) that it might not be plowed. However, I have driven Wind River to 25 to Randal shortly after Memorial Day.
As to the routes near Tacoma, it probably doesn't matter. I've taken 161 more often, simply because I'm more likely to head south on 167 than I5.
