Use of Tukey window and its effect on a graph
The technique that you have described looks similar to the Welch's method for estimation of the power spectral density. Though, it seems that there is an error in your implementation: you should take the squared absolute value of the FFT, instead of the complex value. Such an error might affect your results significantly as the random phase might bring you to a zero mean value of the spectrum. Anyway, most of the languages used for signal processing such as MATLAB and Python contains a ready to use implementation o the Welch's method, so you do not have to mess with the details.
Suggestion: since the variation of the spectrum is due to the user of several windows (variation reduction proportional to N), you might want to overlap the windows. The typical overlap for the common Hann window is 50%.

Window type should not affect the spectrum in a way that you have described. Multiplication in time is equivalent to convolution in the frequency. Therefore, the windowing effect should mostly affect your spectrum leakage. Is your signal stationary? this is a fundamental assumption for using such a technique. 

Actually, as long as I know, the signal that you have described does not contain information in the time domain but only in the frequency domain. In other words, there should be no periodic components with defined phase. Therefore, it is usually processed with a rectangular window (and then there is no need for overlap). 

Using any window reduces the energy as the window's value is smaller than 1. This could be compensated by dividing the result with the window norm/mean value (maybe with the power of two or some other coefficient - depends on the type of the spectrum you want to estimate). Anyway, you don't have to worry about it if you use the implementation by MATLAB which wrappers it all for you.