Does tire pressure drastically affect crosswind stability?

Probably not. But if the max. tire rating allows that much pressure, try it.

But then you are driving a big barn door on wheels. All vehicles that size have some problems in high cross winds. The solution is to stop until those extreme conditions subside.

Might be worth while to have the steering and suspension checked though.

The people I have ridden with that have the most trouble with winds are drivers who over-correct and make the problem worse than it is.