I recommend applying additional potassium (no more than 1# potassium per 1000 Sq Ft) now and in late August per your soil test recommendation.  You should wait until the end of Aug to overseed as the hot, dry conditions of summer are too much for seed to germinate and survive.  I suggest using either #1) a core aerator (lawn will need to be moist) or #2) a vertical mower to cultivate the soil after you have applied starter fertilizer (per soil test recommendation) and 1/4" to 1/2" of organic matter (compost works) to mix them into the soil surface. Then apply seed and lightly rake the lawn to get better seed-to-soil contact.  Where your lawn has really sparse areas, apply clean straw as a sun shield for the soon-to-germinate seedlings.  Irrigate as needed to supplement the fall rains.  By October you should have a nice looking lawn!
Here are links to articles on overseeding/seeding your lawn and more:
Renovating A Lawn for Quality & sustainability-UMN
How to Overseed A Lawn-Iowa
Lawn Renovation & Over-seeding-UMass
Establishing A New Lawn Using Seed (E2910)-MSU
Turfgrass species ID & Cultivar Selection (E2912)-MSU
Core Aeration of Lawns-MSU
Core Aeration of lawns-Illinois
Mulch Leaves into Lawn-MSU
Using Composts to Improve Turf Performance-PSU
Smart Lawns Tip Sheets-MSU