Your breath support may be the issue, as well. This article may be helpful:
Of all the students in a beginning band program, the oboe players usually struggle the most. Even with a good reed and working instrument, without good embouchure and breath support, producing any sort of sound on the oboe is difficult at best. Perhaps the biggest mistake that beginners make is to fight the natural resistance of the instrument. When students first feel the back pressure that results from blowing through the oboe, they instinctively hunch forward, tighten up through the chest and shoulders, drop their head, clamp down on the reed and try to blow harder through the small opening in an attempt to overcome the resistance. However, this only makes the instrument harder to play.
I'm finding a lot of good info on this subject with just simple online searches for "Oboe embouchure".
Be sure to watch this to the end for the reed exercises test. Are you attempting these exercises with the same results? When it comes to practice, nothing is too trivial or elementary...
This is a 2 part blog from the same lady on embouchure.
Specifically regarding register transitions; this article:
It is exceedingly tempting for the beginner to play with a very loose embouchure in the low register (since the tones will sound, although badly) and tighten the embouchure and bite the lips together as he ascends to the upper register. The embouchure should remain very nearly the same from top to bottom of the instrument. Pitch should be controlled as much as possible by stomach muscles, diaphragm muscles and the resultant difference in air pressure. This system is most likely to produce an even tone quality from top to bottom of the instrument.
Watch for air pockets in the cheeks and lips.
A teacher must train his ear to be an accurate and rapid diagnostician
of tonal ills. The exterior appearance of every embouchure will be
somewhat different due to different thicknesses of lips, teeth
formation, etc. By listening, however, it is possible to tell a great
deal about troubles that may not be apparent to the eye. For example,
too much reed in the mouth produces a characteristic ìhonkyî
uncontrolled, blatant sound. Third space C will usually be very sharp,
and other notes may be sharp as well. Not enough reed in the mouth
gives a muffled, stuffy quality. A collapse of the ring of muscles may
produce much the same result since the lip softens and spreads along
the reed and to the tip, restricting the vibrations.