Is it possible to switch from hydrocodone to oxycodone and be in more, rather than less, pain?

Yes. Switching opiates is a bit tricky in terms doses. Also, more opiates can cause a reverse effect with a fancy title - opite induced hyperalgesia. That is to say a nervous system change from the opiate resulting in more pain. Talk to a pain medicine specialist.
Depends on the dose. Oxycodone and hydrocodone are synthetic opioids and belong in the same family as morphine. Oxycodone is about 1.5-2 times more powerful than hydrocodone on a 1:1 basis. However, if the dose conversion is not done properly then patient might experience more pain.