As someone who was part of the layoffs in Feb, I can say that the Sears paperwork for severance is terrible. Conflicting language to start with, but very little help from the HR dept who directed all questions to the 88-sears number (and they're terrible, too).
The basics of the severance are;
You have to fill in and fax the paperwork back in order to be eligible for severance.
Once you start getting severance, if and when you get a new job you are required to tell them within 5 days of starting.
Once you have that new job, you forfeit the rest of the severance pay.
Also - the way it worked in Illinois, was that the 'severance' pay was in fact continuing pay for your position, but you were on a 'leave of absence' which means you couldn't get unemployment until that severance was finished. This goes back to conflicting language, because they called it severance, but ALSO 'leave of absence' in the paperwork. I also could not touch my 401k until the severance period was complete. (I cashed out, needed the money).
It's in your best interests to get the paperwork in and get the severance pay. They won't take out 401k, but they DID continue my medical coverage (not cobra, but regular coverage) for the duration of that severance. It was basically paychecks on time until it was done.
Good luck - I look forward to hearing what departments were most effected by todays events.