Yes, you are allowed to apply again. The important thing is to meet the requirements of the visa, as mentioned here (click to see the evidence needed).
https://www.immigration.govt.nz/new-...-visa#criteria
The application will ask if you have ever been refused a visa before, and it's very important to answer truthfully. You can tell them why you were refused, and the reasons you have told us here are no shame to you - just that you didn't meet the criteria at that time.
To prove that you are a bona fide visitor, you need to show evidence that you have such things as a home, a life, a job and commitments waiting for you back in the USA. For instance, you can show proof of home ownership or rental, and financial records showing that this, and other expense related to living there is ongoing, and hasn't been wound up before your trip. For a job, you can show a contract and pay slips, and something from your employer showing when you are expected back after your holiday. Other aspects of your life could be tickets for something you're going to attend after you get back, a booking for you to go on a trip with friends, you on the rota for doing something at a club, etc. - you know your own activities, so get people you know to supply proof you can show.