Hi,
With the limited time, I would recommend that you focus on the Sabah part of Borneo.
Kota Kinabalu (KK) is the logical entry point. You can do the climb from there (as above replies remarks, be aware that there might be restrictions after the earthquake in early July). Your beach time could then be at the islands just outside KK, the Tunku Abdul Rahman National Park (Manukan, Mamutik, Sapi, and Gaya islands). This is a very easy day trip. Small boats work as ferries from the Jesselton Ferry Terminal, and you can buy tickets for one island or multiple Island tickets. I did Manukan and Mamutik in one day, no problems).
You can then take a convenient 45 minute flight from KK to Sandakan. I paid MYR 79 one way with Malaysia Airlines. The 6 hour bus is an alternative, but since you do not have a lot of time, I would recommend flying at least one way.
In Sandakan, you can arrange a tour along the Kinabatangan for example, or you can go to Sepilok by bus or a 35-50 MYR taxi and see the Sepilok Orangutan Rehabilitation centre, the Sun Bear sanctuary, a rainforest centre, and a probescis monkey sanctuary. I stayed 3 nights in Sepilok and really enjoyed it. A few great accommodation options, both budget and resort type. I stayed at a "dorm" at the Sepilok ForestEdgeResort, and while the dorm rooms (2-3 bed rooms) are a little dated, the facilities, available also to dorm guests, are more resort type (jungle style, not 5-star caribbean style).
From Sandakan, it is also easy to go to the Lankayan Island.
I don't think you will have time to do both the climb and the tour down the Kinabatangan River.
Possible itinerary 6 days:
Day 0: Arriving to KK.
Day 1: Islands outside KK (beaches)
Day 2-3: climbing Mount Konabalu
Day 4: flight from KK-Sandakan (or take the bus and get of at Sepilok) and going to Sepilok. Stay in Sepilok
Day 5: visit the Orangutang centre (do both morning and afternoon feeding). Consider the rainforest centre as well
Day 6: relax at Sepilok and grab a late flight back from Sandakan-KK (or maybe wait until day 7 depending on the time of your flight home).
Day 7: fly out of KK
Remember that it is easy to schedule to much. Flight internally in Sabah are easy to book, so you could wait until you arrive before you schedule anything. Only the climb of Mount Kinabalu needs to be planned.
Hope this helps. I was just there, in May, and Sabah is absolutely beautiful.