Nuclear winter wouldn't be from radioactive dust. It would be from widespread soot from the cities being on fire, creating way dirtier smoke than say widespread forest fires would. The cities would be on fire because they'd be hit by airburst nukes, which cause minimal fallout. Airbursts are far more destructive if you're not hitting a hardened target, which is why they were used in Hiroshima and Nagasaki. There is constructive interference between the initial blast wave from the explosion down to the ground and the reflection of the wave at ground level that makes them far more powerful for wiping out urban areas than a groundburst where a lot of the energy gets wasted going into the ground.
If you're in the north central US, eg Colorado, Nebraska, Montana, the Dakotas, Minnesota, Iowa, or Missouri radioactive dust will be a problem because of the missile silos in Montana, North Dakota, Wyoming, Nebraska, and Colorado that would be hit with groundbursts thus kicking a ton of dust up into the mushroom cloud and giving the radioactive fission products like cesium, iodine, and such something to bind to and be carried by the wind. But fallout wouldn't be anything close to a nationwide problem. It would be for the areas of the north central US that got unlucky with wind direction on the day the missile silos got hit.
If you ever see a nuke go off over land and the mushroom cloud is white there isn't much worry of fallout. If it's dark you know it has a ton of dust and you're completely fucked if the wind is blowing from it and in your direction.
Sad about the fucking idiots in our government / military groundbursting nukes in Nevada and causing that town in Utah to have skyrocketing cancer rates. But that's just not how real nuclear war would be unless you're in the same area as the missile silos.