You always have a choice. The question is do you get severance if you elect not to accept the rebadge. Here, the company has the upper hand because you are guaranteed at least a year of like salary and benefits (based on the contract they sign), so they don't consider it an involuntary termination if you choose not to take the offer. It's considered voluntary. In 2018 we were fortunate only because the VSP was happening at the same time. Maybe things have changed with subsequent rebadges.
Usually the contract is written to guarantee at least a year, but they could negotiate 2 years as well. If you are terminated from the new company < 1 year you would get Vz severance, otherwise you get the new company severance (if they have one). I say usually because the contract can be different than previous rebadge scenarios.
Everyone impacted needs to weigh their options carefully. Much depends on how close you are to retirement and how much you can afford in medical coverage if not 65 yet or what your realistic employment options are in this job market. The answer won't be the same for everyone. At the very least, assuming no severance, accepting the offer would give you a buffer of another year to look for another job.
I've been there, so I know how it feels. However, I can tell you that I know several people who went to Infosys and are still employed after 5+ years and still working on the Vz contract. I personally chose to leave. Good luck.