Here's a version that follows your connectivity, maintains the 0.5% deviation, only splits the three big counties, and generally addresses the negatives.
Tacoma is now in a Pierce-only district so only the remainder of Pierce only makes up 20% of the Olympia district. The Everett district is 63% Snohomish and 37% Kitsap. The remainder of Snohomish is only 37% of the Bellingham district. So, King dominates 3 districts, and Pierce and Snohomish each dominates one district.
This uses your statewide map and adjusts the districts in the Puget Sound area.
Island is shifted to the Pacific Coast-Olympic Peninsula-Olympia district. This reduces the portion from Pierce, to about 10% of the district, the area NW of the Tacoma Narrows.
The northern district comes further south, and now is almost a 50-50 split between the 3 northern counties and Snohomish. It includes Everett and areas to the east and north.
The transcade district is now split: Snohomish 56%, Chelan-Douglas 17%, King 16%, Pierce 11%, with Edmonds and the Seattle suburbs south of Everett dominating the district.
The split in King County is:
(1) Seattle, Shoreline, and a few small towns across Lake Washington near Medina.
(2) Southern suburbs as far east as Tukwilla, Kent, and Auburn, plus Vashon Island, and Kitsap County. About 63% of the district is in King County, and it includes 23% of the county.
(3) Eastern suburbs across Lake Washington, including Redmond, Bellevue, and Renton, as far east as Sammamish and Issaquah, as far south as Covington and Maple Valley.
(3) The eastern part of the county, beginning with Duvall, Carnation, Fall City, Mirromont, Hobart, Black Diamond, Lake Morton, Berryhill, and Lea Hill. This is about 16% of the entire district, 6% of the county.
The split in Pierce County:
(1) Gig Harbor area, NW of Tacoma Narrows, about 10% of district, and 8% of the county.
(2) Tacoma and southern and eastern suburbs, as far east as Sumner, Puyallup, Orting, and Graham. 83% of the county.
(3) Eastern part of county, with most in the Bonney Lake, Prairie Ridge SSE of Auburn, about 11% of the district, and 9% of the county.
This moves the 3-way split from Snohomish to Pierce. The split in Pierce is pretty reasonable with most of the built-up area in the Tacoma district.
Since the transportation link is US 2 and Stevens Pass, I think Snohomish is a better pairing than King for Chelan and Douglas. The northern Seattle suburbs are a bit more distant than those across Lake Washington, and I think it is reasonable to include Everett in the northern district.
I also like the southern suburbs paired with Kitsap. The Edmonds-Kingston ferry appears to be a bit more oriented toward being a link to the Port Angeles and Port Townsend areas. A small flaw is that the terminus of the southern ferry from Kitsap and Vashon Island is actually in the southern tip of Seattle, but only about 1.5 miles from White Center.