Take your results to the county extension office, double check with them --- that seems like lot of lime. Adjustment of pH of the soil is important to correct before you start any fertilizer program....

Did the soil test also tell you how much fertilizer to apply?

As the temps get hotter, ( like now) grass will grow much slower and eventually go dormant. If you want to use chemicals to kill the weeds - Weed and feed may be helpful now. Apply in am when dew is on the grass, and no chance of rain in 24 hrs. Or spot spray weeds with a weed killer ( like weed b gone), Then fertilize in the fall with a fall fertilizer mix ( lower in N)

Next spring ( again if you want to use chemicals) - try applying a crab grass preventer ( usually comes with fertilizer) as the days start to warm and before the forsythia blooms.