Oboe/Intonation on top line F (standard fingering!)
When fingering F at the top of the staff using either the right hand F lever or the left hand F key, not always, though far more often than I would like, I get a "volunteer" F#. This is not an out-of-tune F, but a genuine F#. If I reduce the air it will often return to F, albeit not a pleasant one, and pressing down white knuckle style with my RH index finger will correct it, not practical for technical passages! I am just staring with a conservatoire system oboe, graduating from a Boosey & Hawkes thumbplate which only plays the notes I actually finger. Is it the horn or me that's in need of repair?
Wobbly Fs are a common failing of some oboes. The problem is caused by tone holes for E and F being very close together. A well designed oboe minimises this potential failing. Stiffer reeds will help enormously as will a relaxed embouchure. However some oboes are very prone and almost incurable.
Unless of course there is a problem with the adjustment of the first finger RH lever that closes the G sharp key if you had opened it. The first finger being used for F sharp would then automatically close the G sharp key. If the adjustment is too "strong" then the pad under that plate dosn't close properly. So test it by turning the bottom joint to the right so that the lever no longer impinges on the G sharp key.
Now try the F natural and see if the symptoms have eased or gone altogether. If they have you need to adjust that connection. If there is a screw that's easy but if not a bit of bending will be necessary or sanding down the little piece of cork on the top of the G sharp key to make it less high
From what you say about "white knuckles" it sounds to me like this is the problem. This connection can be quite tricky to get right so I hope you have a screwed adjuster to help.