That the time is an exp(i Et) is true only in stationery states. What will happen when the wave equation isn't separable? It may not simply get absorbed while normalizing anymore.
The wave function solution psi is a function of time and position. Hence the integral of its square over all x will, in general, give a function of time. To normalize this, we must multiply with the inverse of the function. Therefore it seems that the normalization constant does not remain...