LNG exchangers usually have a tight temperature approach so that the hot stream temperature stays very close to the cold stream temperature for most of the length. The simplifying assumption that 'heat capacity is constant unless there is a phase change' is not good enough! The heating curves for your streams are not exactly linear. Since the two heat curves are close together, there can be an internal pinch that is not obvious from looking at the inlet and outlet temperatures. I do not know the details of your problem, but you should calculate the heat exchanger using many internal regions. Some modeling software allows the user to specify a minimum internal temperature approach (MITA) which is extremely handy. View the plots of your heat curves on the same graph to get a better feel for what is happening with your calculations.