The US had numerous contingency plans for "continuity of government" in the case of nuclear exchange over the years. They vary depending on the scenario imagined but most imagine that some remnant of the original government will survive and it will restore order and begin rebuilding fairly shortly. Some of these are very optimistic. It is worth noting that none of them involve staying in underground installations for long periods of time with the exception of the possibility of a "shadow government" having that capability (and this was more about it surviving the attack than living under there continuously) at places like Raven Rock. Contrary to popular belief a nuclear war would not require living for decades underground; after a few weeks the levels of radioactivity would decay from "acutely dangerous" to "a chronic threat" and by "chronic threat" you mean an increased long-term cancer risk. (To put it into perspective, the lifetime cancer mortality for Americans is currently 20% or so. Adding, say, 10% to that would certainly be a bummer, but life would still go on.)

Cold War estimates predicted that "only" some tens of millions of Americans would die in a nuclear exchange with the USSR, and while that is probably very optimistic on behalf of the government, even if we increased that to, say, 100 million, that is still "only" 1/3rd of the people. That would be, to be sure, the destruction of whatever had previously been considered the United States, but it is not the total extinction that people tend to imagine. (I am not trying to "lessen" this — personally I think trying to imagine every third person you know dead is actually far more disturbing than just imagining everyone being dead.)

Now it should be said that even within the government there were people (often the President) who looked at the "optimistic" estimates and said, this is bonkers. To lose 30 million people in one go, even if that is "only" 10% of the US population, would unthinkable and would be the end of the nation as it was defined. (Consider how tenuous things were and have been after 9/11, when only 1/10,000th of that many people died, and mostly those of only two cities.) In the 1970s, even the public-consumption versions of these estimates included a somewhat dark acknowledgment that it was possible that, despite any plans and hopes, the result would be a downward spiral into "chaos and anarchy".

As for nuclear winter, the US government never really took it into its calculations because it never had strong confidence that it was real (and took refuge in that uncertainty). Furthermore if the nuclear winter hypothesis is true, it indicates that the survivability is far lower than even the pessimistic estimates would have it, because food shortages would be rampant as a result.

I am generalizing for many things here but if you are interested in these matters, this is essentially the history of civil defense, and there are several books on this subject (e.g. Kenneth Rose, One Nation Underground). On the US government response to nuclear winter, see Badash, A Nuclear Winter's Tale. There is a recent book out on Raven Rock and contingency plans by Garrett Graff but I have not had an opportunity to look at it yet.