The sentence "The highway travels north from Interstate 705 (I-705) in Tacoma along the Puget Sound through south King County, serving the cities of Federal Way, Des Moines, and Burien west of Seattle–Tacoma International Airport to SR 99 south of downtown Seattle." needs to be reworded. I would suggest splitting into two sentences with the first sentence mentioning the termini and the second sentence mentioning the cities it passes through.
Split
The sentence "SR 509 continues north onto Marine View Drive through the city of Normandy Park and into SeaTac,[24][25] where it turns southeast onto Amabaum Boulevard and Des Moines Memorial Drive to a trumpet interchange, the southern terminus of a limited-access freeway section of SR 509" needs to be split.
Split
The sentence "SR 509 passes Highline High School and intersects Southwest 148th Street in a partial cloverleaf interchange, forming the western terminus of SR 518,[29][30] before a half-diamond interchange with South 146th Street." sounds awkward.
Re-worded
You use the verb "intersects" a lot in the route description. Can you please try to vary the wording a bit?