I went to Fiji in February 2007. Our group went on the
Nai'a Live aboard.
The logistics were pretty easy. There were several non-stop flights from LAX to Nadi, Fiji. (For me, the best option was Air New Zealand. I am glad that I chose them, the flight was great and compared to typical North American airlines, it was very spacious and comfortable even in economy.) Once in Fiji, our group got a "day room" at an airport hotel to freshen up and relax in until the transport from the Nai'a picked us up. From the airport, it was about a 30 minute drive to the dock.
Once on the boat, we settled in and soon got underway. The Nai'a has undergone a refit since I was on her, and the
cabins look very nice. I think that even for a "Bucket List Trip", you would be able to find one that matches your needs.
I can't really speak to what the conditions are like in the May - September. I was there in February and there were some wild currents. When I say wild, I don't just mean that they were strong. They also changed direction as we descended. On at least one dive (E6), the Zodiac dropped us off "upstream" of the dive site anticipating a nice easy drift onto the site. The current however had other plans and at about 30 feet it changed direction by about 90 degrees and we were "blown off the site". (Never did get to see E6.) We also had to find a nice sheltered lagoon one night surrounded by mountains after as a cyclone blew overhead. (We were sheltered from the wind, but had a
lot of rain that night.
We did do a shark dive, and there were a lot (25-30) sharks there. They were mostly Grey Reef Sharks, but there might have been a Bull or two in the mix, I honestly don't remember. IIRC, it was at the "Cabbage Patch" so if you are looking specifically for Bull Sharks, you might contact the Nai'a and ask them.
You wanted to know about when to leave and some of the logistics. I'm not sure where Princeton TX is (I lived in San Angelo for a few years in the late 1980s / early 90s.), but I assume that you will be flying out of either DFW or IAH. From there, your flight will likely connect through LAX for a "red-eye" to Nadi. When I went, the flight out of LAX departed at around 10pm Thursday evening. A couple hours later, it was midnight and now Friday. A few hours after that, we crossed the International Dateline, and it was now Saturday. The flight (which took off from LAX at about 10pm Thursday evening) landed in Nadi Fiji at about 6am Saturday morning. Coming home, the International Date Line will have the opposite effect. I gained back the day that was lost on the way going to Fiji and I landed in LAX "before" I departed Fiji. (In fact, I took the redeye out of Fiji on 10 Feb to LAX and then connected to the redeye from LAX home also on 10 Feb.) This is really just an administrative issue, but it is something that you have to be aware of when booking your flights. If you are hoping to get to Fiji a day or two early, then I would look at departing TX on Tuesday or Wednesday if your schedule has a Saturday departure. I don't know if the Nai'a still picks up at airport hotels, but they did for us when I went.
Edited to add: If you are looking at flying out a few days early to get a head start on the jet-lag recovery, one option (especially if you fly Air New Zealand) would be to book a flight connecting through Aukland or Christchurch and spend a day or two in New Zealand on your way to Fiji.
I hope that helps a little and that I have given you some insight into how things might play out.
Have a great trip no matter what you decide to do.