Straw is normally used only when you are seeding into bare soil. Not suited to seeding into existing grass. Foe cool-season grass, seed in late summer or early fall for best results. When temp comes down to about 85. Highest soil temp is about late August...that is when germination will be the quickest...provided you have good soil moisture. It is a bit of a guess predicting when rains will happen in fall. Try to seed 8 weeks before frost.
Warm season grass--like Bermuda and zoysia should be seeded in late spring about 8 weeks after frost. Temps above 85; they love heat, but still need water.