The US is far more complex than that with some areas being significantly more Democratic or Republican. Also candidate also choses a point on the political landscape and often needs to beat people within their own party before going after the general election. So there are plenty of Democrats that are far more conservative than elected republicans and vice versa.
This suggests that each race may have a seperate Nash equilibriam based on the canidates and the electrate.
From what I heard gerrymandering means ~2/3 of general elections are fairly uninteresting. But, you still see fairly wide swings in public sentiment leading to a fair amount of churn in public office. Also, in extreme cases people can treat the primary as if it was the general election.
This suggests that each race may have a seperate Nash equilibriam based on the canidates and the electrate.