Paying Costs you Cannot Afford
No, in these cases you are welcome to choose the "discard 3 cards" option and merrily not do anything.

These effects of Crises are not really COSTS per se - in most games costs are non-optional, so your reasoning about being forced to do the other choice makes some sense if you think of these as costs.

But they really aren't costs, they're just the effects of that Crisis, and you're welcome to choose the one that impacts you the least, and execute as much of it as you're able to.