How to properly transition from one contractor to another

I have been working at Technion in cleaning for 5 years now — 4 hours in the morning and 2 hours in the evening. All this time I have not been on vacation, there are no employment conditions, I received money for the morning in cash, and for the evening — by check. Now the contractor at Technion is changing. I would like to stay there because it's close to home, not many hours, not so difficult. But the problem is that the previous contractor will not give me a letter of dismissal — either I have to resign myself, or he will transfer me to other sites — far away, harder, more hours, etc. The new contractor is ready to accept me, but only if I leave the old contractor. What should I do? How to properly resign to receive severance pay, and am I entitled to them at all?