The Wenaha-Tucannon Wilderness should be just fine by July 1. Current "Snotel" data (I looked yesterday) shows the snow gone at 5700 feet in that area (Oregon Butte, the high point, is 6500). Generally the higher trails (like Oregon Butte) are accessible by mid June, even in high snow years. It's better to go when there are still scattered snowdrifts left at higher elevations than to go later on when the springs have dried up! However, it will be really hot down in the Wenaha River Canyon. You also need to keep an eye out for poison ivy (yes, ivy, not oak) and rattlesnakes while down in the canyon.
Another possibility would be a loop on the Oregon side of the Columbia River Gorge (my "back yard"). By July 1, the snow in the upper areas (only 4,000-4,500 feet, but north-facing slope and close to Mt. Hood) should have melted out. You could go via the Gorton Creek or Herman Creek trails (the former has more viewpoints but is more strenuous) to Wahtum Lake and return via the spectacular Eagle Creek trail (you might want to avoid the latter on weekends and/or holidays, though--it's very popular with dayhikers, hence crowded). If you do this, park as close as you can to the campground host at either Herman Creek or Eagle Creek campgrounds, or park at the Eagle Creek Fish Hatchery lot (definitely not the Eagle Creek trailhead lot), and leave nothing (of value or otherwise) in your car. There's a trail (Gorge Trail #400) connecting the two trailheads.
A third possibility would be the eastern end of the Pasayten Wilderness in northern WA, in the Horseshoe Basin area. It's in the rainshadow of the North Cascades so melts out earlier. If you're using Doug Lorain's "Backpacking Washington," it's listed as the "Eastern Pasayten Loop." However, most people go in from the Iron Gate trailhead to Horseshoe Basin, although the "string" on the "balloon" is about the same length as the Thirtymile Campground entrance suggested by Lorain. From Iron Gate ther's less elevation gain and, I believe, less "cow country." Again, there may be snowbanks but not solid snow. This is the trip I personally would pick.
A fourth possibility is the awesome Olympic National Park wilderness coast. It's certainly a lot different from either Richland or Indiana! It is, however, very strenuous (up and down primitive ladders and steep muddy trails when going over headlands) and requires a lot of attention to tide tables. You can find details on the ONP website (www.nps.gov) or the above-mentioned Doug Lorain book. You'll need campsite reservations for the Sand Point-Cape Alava section, and those may be hard to get on the July 4 weekend. There are, I understand, private shuttle services available, since it's a one-way hike.
I would say that any such loops in the higher Cascades, Olympics or Wallowas will still be pretty snowy.
You might want to post a query on www.nwhikers.net (Seattle) or www.portlandhikers.org.
I lived in Richland for 9 years, but fortunately was able to move to Portland when I left. I did enjoy my time there.