Where are you flying from? If you're starting in the EU or UK you have protection in law to be rerouted.

I also suspect that the agent was just getting things wrong. Unless a route is completely cancelled, airlines are usually fine to shift you to another flight that they operate a day or so either way. Getting them to reroute you on another airline is a harder ask as that costs them money but if you're flying from the EU or UK you are protected.

Key things:
Don't panic, you have time to sort this out for January
Don't accept a refund, as at that point you become a former customer and have no right to anything (at least not easily)
Get some more input from this thread before calling them again.