The process of exporting PDF to word remains the same on trial as well.
You get an option to continue to use it as trial or sign in to license the product.
You could choose the first option.
Trial is valid for 30 days.
I hope your Acrobat trial has not ended and that is why it prompting you again to sign in.
I wonder if you might still be running Acrobat Reader, even though you also have installed the trial for Acrobat. Run each one in turn and be sure you can see the difference.