Can we use both FFT and PSD to obtain a random signal's frequency domain representation?
As Marcus Müller said in the comments, PSD is an output (an "answer"), while FFT is a calculation ("a way of getting an answer"). There are many ways to calculate the PSD, and they are all "estimates" since your input signal is noisy and uncertain.

More importantly, if you take the FFT of your raw signal, then you will get the amplitude spectral density. What you really want is the power spectral density -- the PSD. The most basic way to estimate a PSD is to simply take the FFT of the autocorrelation function of your signal. But that PSD estimate will be very noisy and uncertain (since your input signal is noisy and uncertain). If you add more sample points you just end up with another noisy and uncertain estimate, but one that has much higher frequency resolution.

There are a lot of ways to quiet down this PSD estimate, but spectral analysis of noisy signals is a tremendously involved subject, with entire books and graduate-level engineering courses devoted to it. You can find a good introduction in this tutorial on dsprelated.com. If you Google the phrase "spectral estimation of noisy signals" you will find a bunch of other resources including entire books. (I think they may be violating copyright law by putting those online, so I'm not sharing the link here.)