Yes, season ticket holders get first access starting with full season holders and then partial season ticket holders (40 and 20 game plans)
The White Sox will then announce a specific date and time that tickets go on sale to the public. Use a computer as opposed to a phone (faster) and try to get multiple family/friends trying, as they go fast. Also, not sure if they still sell tickets over the phone for postseason but worth a shot doing phone and computer at same time if they do.
After they are sold out you will have plenty available on the aftermarket sites. Like you said these will be well over face value.
Not sure what pricing will look like as it's been over a decade. I can look up what other teams have been going for in recent years which may give a decent ballpark estimate.