FFT analysis of binned data of different length
Let's say you have several data sets $x_i$ and $N_i$ is the length of the $i$th data set.

First you're right that the amplitude of the FFT output scales with its length. To normalize all data sets they have to be divided by $N_i$. (Note that depending on the implementation of the FFT there might be a scaling involved already, check the Doc or see this question)

To combine all data sets you should append $(\max_i N_i) - N_i$ zeros to every data set before taking the FFT. The frequency axes are then aligned correctly and you can calculate the average of all FFT outputs.

Btw, I would plot the absolute value of the FFT output, not the real part. And sometimes I find it useful to "smooth" the spectrum by averaging every $M$ neighbouring FFT bins. Last, I have assumed that the data of every set has been acquired at equidistant time steps, otherwise the FFT isn't applicable at all.