The job process from post to hire takes an average of three months (mid April), but maybe longer. Why not try and negotiate a later start date? Have a new job and severance? $$$
If you think they are trying to screw you out of the severance...DON'T look for a job. Just sit around until you can get your lump sum check....and then HOPE and PRAY you get a job before your severance runs out. All the smart people that started looking early will have already filled the jobs that you are qualified for. Think people....a lot of folks with similar skill sets looking for a few jobs. Take advantage of learning this news sooner versus later!