I mean, yeah, 1000 people is enough assuming there's no sampling bias. But if you've got sampling bias, increasing the sampling size won't actually help you. The issue you're talking about is unrelated to how many people you talk to.

Your own suggestion of splitting up the respondents by state would itself introduce sampling bias, way over sampling low population states and way under sampling high population states. The survey was interested in the opinions of the nation as a whole, so arbitrary binning by states would be a big mistake. You want your sampling procedure to have equal change of returning a response from any random person in the nation. With a sample size of 1000, you're not going to have much random-induced bias for one location or another, aside from population density, which is fine because the survey is about USA people and not people in sub-USA locations.