State tax filing

I have a similar situation.  I am a home office based employee working and living in NJ but the company I worked for has its headquarters in CT.  The company withheld NJ taxes only.  Do I need to file a non-resident CT return?

 

In addition to my current company in CT, I also received severance payments from another CT-based company while living in NJ.  That company only withheld CT taxes (no NJ taxes).  How should I file those earnings if I was not working for that company any longer, however was receiving severance payments reported as income from a CT-based company?