As voyager1 says, they're good until at least September, and they'll get used up well in advance of then in Taiwan.
The thing with the AZ vaccines was Japan ordered them before the blood clot stories came to light. There's only a tiny risk of getting a blood clot, but the Japanese authorities are being very cautious so there are going to be lots of spares.
In the UK AZ was offered to everyone 40 and over, so Taiwan could easily do the same. Pretty much all my relatives 40 and above had AZ and reported no ill effects.
Unfortunately corruption in China has barely improved over the last decade.
(China's score has improved by 3 points since 2012.)
Xi, like some of his predecessors, has not really sought to deal with corruption on a wide scale, rather appearing to prefer using it as a weapon to go after his opponents and other people deemed too much of a threat.
Every CCP leader has had anti-corruption drives, but until they clean out their own house (i.e. the Standing Committee, Politburo, families of top politicians, the princelings, etc) I'm not sure much is going to change. At the very least the government needs to significantly increase public-sector wages so that there's less "need" for people towards the bottom topping their wages up.