I am LGBTQ, so it seems like Auckland or Wellington will be my best bet.

Yes, good call. You'll be welcomed, but just remember everywhere has bigots.

If I’m in a big city, I would prefer not to have a car.

I know many people in Wellington (where I live) who have no car. Public transport is pretty good, and for day to day life you don't need a car so long as you're somewhere serviced by public transport (preferably with a direct route between you and where you work). As I understand it, Auckland public transport has taken great strides in the last decade or two and so should be similarly easy to get around - but the city is spread over a huge area so consider where you are working and make sure there's a transport route that doesn't take two hours and three bus changes to get there! If you have work lined up then when looking for housing just throw it into Google Maps and see what the available routes look like.

I’m a nurse, and it looks like salaries are good but not great (between 70-90k/year is what I seem to be finding).

I believe there's a lot of available movement from that - for example, as a Nurse Practitioner with a lot of experience you may be able to double that range. That range might be appropriate for you now, but the top range for experienced nurses will certainly be a lot more than that, it's not that nurse salaries cap out at $90k.

Is NZ easy to settle into?

As NZ born I can't answer that, but know that over 25% of NZ's population was not born here. You'll be surrounded by other immigrants, which is especially true in larger cities.

I have a nurse cousin who said the US doctors struggle with the autonomy of NZ nurses. Apparently in the US if the doctor prescribes a patient some paracetamol (you might also have to learn new names for the same drugs), then the nurses will make sure they give it to them. In NZ, the nurse might ask the patient how they are doing, and give them all, some, or none of the paracetamol (or whichever drug) based on how well their pain is being managed. It's apparently quite a change for US doctors to understand that in NZ nurses can use their own judgement.

In terms of settling in, I have heard it can also be hard to form new social circles. Other parts of Lemmy have posts on this from time to time, I think the general consensus is to join clubs to meet people with common interests. Often immigrants will form social circles with other immigrants due to the shared experience.

Am I going to be broke all the time (I also have a cat I am bringing unless I am physically unable to)?

Realistically that salary should be enough, even for Auckland/Wellington, but the biggest single expense is likely to be housing and how much you spend there is likely to determine the outcome to this question. Also be aware that housing can be cold and damp in NZ, so it might be worth feeling poor to have a warmer house. I would expect that in Auckland or Wellington, $70k-$90k is not going to make you feel rich but should be enough to live comfortably if you have no debt.

I'd also suggest buying electronics or other big purchases before you come (or at least researching what they cost to buy locally). Imported goods can be quite expensive here.

I did some research prior to applying, but I find more and more opposing views on where the country is headed. I guess I am just looking for some folks that had a similar situation to see how it worked out for them.

NZ is not immune to the same things that are affecting other developed countries. However, I think we are a long way from the government kidnapping people off the street and sending them to prisons in El Salvador without trial, so I think you'll find it a nice place to live if you can build up your social circle. You can also stick it out to get citizenship then jump to Australia if you find the salaries not enticing enough in NZ.

Just as a final comment, I'd make sure you actually have a job lined up if at all possible. I have read recently that the NZ heath department used to hire 90% or more of all new grad nurses in NZ, and that has dropped in the past couple of years to more like 50%. To me that sounds like it's getting hard to find a nursing job, though you might not struggle if you have enough experience. Best to actually line something up though if you can.