The time of year you've selected is as good as it gets. Rockies to east coast, August is still summer, often the hottest part with periodic summer storm systems, and as the air mass boundaries move south in September and October, progressively cooler in the North, though summer conditions may prevail well into September while Tornado Alley shifts back down from the Great Lakes region to the southern plains and mid-south.
Depends on how far north you want to cross, but maybe beginning of September through end of November might have more moderate weather, but there is some risk of early winter storms in the Rockies and northern Plains in September, and at high altitudes anywhere, and upper midwest, central plains, even southern plains after end of October.
Depends on what you consider bad weather, what you are trying to avoid. I think for most of the middle of the country, September and October are the best season for outdoor living, and November if we get lucky. But we pick our days based on weather forecasts, go when it is good, stay home when it gets unseasonable or even seasonably stormy. That's not the same as a long trip with "must travel through here now" imperatives.
Depends a whole lot also where on the east coast. September and October are very different from Maine to Florida, and by October can be. very different Montana and Dakotas vs Gulf Coast and Texas.