Vallis & Sagitaria can survive in cold water down to 3 or 4 C. Most swordplants and crypts need to be kept at temps above 20C, however some swords will tolerate 3-4C as well. Most other plants are tolerant of cool water around 10-15C or above.
If you plants are rotting it could be you have marsh plants and not true aquatic plants. Marsh plants are able to stand upright when removed from water, whereas true aquatics will usually fall over when taken out of water. Hygrophillas, crypts & swords are the exception. They will grow under water and above it in a marsh setting.
Can you tell us about your tank, how much light you have, any plant foods you put in, PH and so on?