I am a transplanted Dallas Texas boy that has lived in Honolulu for 19 yrs. I can give some local insight as I am considered Kama Aina now. Honolulu is a big bustling city with lots and lots of people. It is not the ideal destination unless you are willing to be trapped in Waikiki for the duration, basically hemmoraging money after about 3 days. Yes there are sights to see, but they are highly tourist oriented and traffic is a nightmare on most days. Agreed, the Mormon influence is very strong here. The Polynesian cultural center is mormon owned and operated, it is also a bit of a bore.
I would rather recommend Kauai. It is slow, laid back, staggeringly beautiful, and much more natural in its approach to life. My wife and I get there as often as I can. I am not gay, however, it is very gay friendly as is all of Hawaii.
The "Mahu" as gays are affectionately called are and were revered figures in Hawaiian culture. They were different and special. That said, Kauai offers superb beaches, the Napali Coast, the Waimea Canyon, zipline tours, and rafting tours especially are great. We took a tubing tour with Hawaiian adventure tours which was laid back and properly and safely run. It cost around 165 for a couple but was worth the trip and the experience.
A place I have stayed that was quaint, affordable, beautiful and family run was and is the Kauai inn in Nawiliwili Harbor area of Lehui. Highly recommended if you want quiet, and romantic. A noted restaurant there if you like mexican food is Mariachi's. Mexican food is a toss up in Hawaii, I loved the place being an old Texas boy.
I cannot recommend a stay in Kauai enough.