In DC, you can't apply for unemployment until your severance runs out. I received 10 weeks' pay as severance, for example, and I received it in one lump sum. I had to wait ten weeks past my last day on the job to apply for UE.
2nd point: the threshhold in DC is VERY low. You can basically work at minimum wage for 8 hours a week and put yourself out of eligibility.
Conclusion: Only do the hourly contract work during your severance period. Once you're ready to apply for UE, you have to stop working altogether or you will lose it. I know. It's a profoundly stupid system.
Caveat: I don't know if having part-time contract wages on your record precludes you receiving benefits based on your former full-time salary. It might. So you should really talk to someone. I used to be able to reach someone at the DC UE office in the mid-mornings. I'd have to sit on hold for 5-10 minutes, but it wasn't bad.