Okay, my turn to step in and you can thank John for letting me know, otherwise I would have found this out a month after you left.
You know the drill with the ITT offices located on every base, right? That's where they sell discounted tickets to expensive events for military folks. Okay, so if you want the real local culture stuff, I would highly recommend two places, which aren't too far from Hickam - Bishop Museum, which has everything about Hawaiian history and culture (and the skeleton of a giant Sperm Whale hanging in the middle of the place), and Iolani Palace, which was restored to the way things were when the Hawaiian Royalty were living in lavishness at the expense of the people before the US Marines took over the islands and white businessmen started getting rich at the expense of the people. Okay, so I thought of two more places - If you are familiar with Ft DeRussy (since you mentioned Waikiki), there is a War Museum on the grounds (look for the Japanese WWII tank parked outside. Finally, this is harder to find, but worth going to, and that is the Plantation Village in Waipahu (not to be confused with Dole Plantation in Wahiawa). The recreated the different plantation camps which the imported workers lived at during the days of paid slave labor. It was a favorite of mine because when I was spending time with my grandparents on the Big Island, things were still somewhat like what that village was like.
As for shaving stuff, not much on Oahu. I heard an AOS shop opened up at Ala Moana Shopping Center, but I never been there. Hawaiian Shaving Company is based on the Big Island, and I believe they don't have a supply chain for Hawaii places except for this one small local artisan shop in Hilo.
Hope this gives you some ideas.
Wait, one more, just to make sure - You mentioned Pearl Harbor. Did you cross over to Ford Isalnd and see the Missouri and the Flight Museum? Most folks miss the Flight Museum.