I live in NJ but my home office is in VT. We were recently acquired by a company from CA. Who do I pay taxes to and where do I need to file?

The question is - where do you work? If your work location is CT, then you do not have to file or pay taxes to any other state but CT. The location of the home office or corporate HQ is immaterial.

I would advise that you contact the payroll department and verify that you have a valid W4 on file that identifies CT as your residence and work locations, and also verify that they are properly withholding CT income tax - but not CA nor VT - from your pay check.