6 nights on the Big Island and 3 nights on Oahu. The reason for such a split is: 1). Big Island is big and you need time to explore 2). Oahu as you said is crowed (really only Waikiki) 3). You can hit the highlites on Oahu over your short time there and still have time to relax. One recommendation I have, do one activity a day. It could be 2 hours or a 6 hour activity but only do one. What is the point of traveling many hours, getting stressed out and not enjoying yourself. One example would be go to Pearl Harbor early, then the USS Missouri and maybe the Submarine Museum one day. It would take about six hours but you could be back at the hotels pool by 2 pm.
I don't know your budget or plans but I would consider staying at a condo on the Big Island for all or part of your stay. You could stay all 6 nights in a condo or you could do a luxury hotel the first 3 nights and then a condo the last three. Alot of my reasoning for this is you can do your laundry at your leisure thus having to bring less on your trip. Think about it. My experience while staying in a hotel and laundry is that you need to get up early to wash and dry your clothing because washer space/dryer is very limited. In a condo, you can wash clothes at your convience. While on Oahu, you could stay in a luxury hotel for the final three nights.
As for things to do, sit down and decide before you go. There is so much to do. I could offer something for you to do, you do it and you hated it.