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Unless you had an agreement to the contrary, you were and are an "at will" employee which means the terms of your employment can be changed at any time. This means that your employer can give you a new contract, which whatever terms they want, and condition your further employment on you signing it. You are free to negotiate, but ultimately if you fail to reach an agreement or if your employer doesn't want to negotiate they have no obligation to continue employing you.
Your employer cannot condition payment for already earned wages on you signing a new contract. You could likely rescind such contract on the ground that you signed it under duress if that happens. However, you cannot unilaterally change your pay but changing the contract. In order for any pay raise to be valid, your employer has to agree to it. Even then, since your employment is at will, they can decide to reduce your pay to its previous level at any time.
As far as unemployment goes, you are eligible for unemployment if the EDD finds that you are unemployed "through no fault of your own." If you are offered a comparable job with the new company and refuse it, you will not be unemployed through no fault of your own since you made the voluntary decision to refuse work. If the terms of the new job are substantially worse than your existing job, though, then this would not be an offer of comparable or "suitable" work and you would still be eligible. Since you are clearly being offered a job with the new company, your entitlement to UI benefits will simply come down to whether the new job is suitable. The EDD has held that if your rate of pay is no more than 30% less and if the job responsibilities are at least roughly equal, the new job will be deemed "suitable" and refusal to accept it will be grounds for UI disqualification.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.