How can I prove stability of a biquad filter with non-zero initial conditions
Tricky problem. Not sure I can answer this but here are a few pointers:


Direct Form II is the worst biquad for audio processing. The transfer function between the input and the state is given just by the poles. The gain can be really large, I have seen gains of in excess of a 100 dB for reasonable audio filters.
This makes real time switching of Direct Form II very tricky. The magnitude of your state can change 10s of dB with seemingly small overall filter changes. This creates massive clicks and pops and occasionally unstable filters.
Much better choices are Direct Form I or transposed Form II. For these filter types the state variables are always in the same order of magnitude than input and output. Much less potential for loud pops and clicks or instability
There is not "sure fire" way of updating an IIR filter in real time without the potential for audible artifacts. Some potential methods are running two filters in parallel and slowly cross fading the output or updating pole and zero locations really slowly with very small steps at the time
If you must update in real time, make sure that you don't update one coefficient at a time but ideally both poles and zeros in one go. At least do the poles all at ones (i.e. always update a1 and a2 at the same time)
Update lower gain first. If the pole section of the new filter has lower gain, update the poles first and then the zeros. If the pole section of the filter has higher gain, update zeros first and then poles
If you need to stick with Direct Form II, it's probably best to zero out the state during update. This will still create an artifact but it's probably less than if you don't and at least it guarantees stability.