Peter D'Amato, in his book The Savage Garden
, has this to say about "subtropical" rosetted sundews:
With a very few exceptions, this class of Drosera grows throughout the year without any dormancy requirements. While most grow in frost-free climates, many are tolerant of brief freezes into the twenties. The rosettes will die during these light frosts, but usually new plants come up from the thick, pretected roots. For best results, grow them in frost-free environments.
Don't worry about the dead leaves on the Spatulata. They may simply be old leaves (you can carefully cut them off if you want to) or leaves that are being sacrificed by the plant due to shock of transplanting or growing environment, which is a common occurrence.
Good luck and best wishes.