Is it true? Can I go to the airport/Spirit counter and get a better low price for a ticket?

Domestic tickets are far cheaper than international ones; you can indeed get a cheap flight from, say, NJ to Florida for less than 60 dollars. U.S airlines compete with each-other, especially now since their reputations have been slowly diminishing from public backlash, making the price of their tickets quite cheap, so yeah, take advantage. Hope I have been helpful!