You can always call the clearing house and ask them not to assign to your account but to dump the bag on some other unlucky person. But I heard you need to know someone in congress.
Lol.... Even if you know someone at the OCC, they do the random assignment per broker. They have no information about individual client/trader. They wouldn't be able to not assign a particular trader even if they wanted to.
Can't avoid assignment, but if the issue is not losing the particular lot(s) of stock there is a process that most brokers refer to as same-day substitution. Essentially you trade a "cash settlement" and deliver that against the exercise. Cumbersome, costly and many brokers don't want to facilitate it anymore. So ask your broker now before you need it and understand what is required
What happens if your account doesn't have enough margin to hold the shares? Do they exercise and immediately liquidate? Seems like a roundabout way to do things.
You have to be specific, ie. exact, as there are multiple possibilities. Which case exactly do you mean? Option buyer / seller, option type Call / Put ? Sorry, I'm not an expert on this specific scenario as it depends also on the broker. But maybe someone else can answer.