Hi, woodwind technician here.
Please only disassemble your instrument if you feel comfortable doing so and know your instrument well. Pick up a screw board from Music Medic or make your own to avoid losing parts.
Additionally, if you’re a regular player of your wood horn, oil at your discretion. Regular playing keeps the wood hydrated, and wood will only accept moisture as it needs. You just need to make sure to clean it regularly.
If you want to test it, brush the ends of your tenons clean. Add a dollop of oil to the cross grains of one socket. Let it sit for an hour. If it absorbs quickly, yes, go ahead and oil. If not, you likely don’t need to oil.
Be sure to wipe the instrument thoroughly after the oil is done leaching.
I highly recommend Naylor’s bore oil if you’re looking for something quality. Also, oil your keywork if you’ve disassembled (after cleaning your hinge rods/screws). I recommend Hetman’s.