FFT on I/Q-data Vs just I-data?

I'm trying to obtain a frequency analysis of a signal in the 40Khz-50Khz band. To do this I first mix it with a 40Khz square wave LO & 4th order elliptic low-pass filter. Then feed the output to a single ended ADC sampling at 20Ksps on a microcontroller programmed with a DFT algorithm.

HOWEVER, i never seem to get totally consistent results on the spectrum. My question is this - would it make a difference if I mixed/ downconverted it to get I/Q signals & then do a complex DFT ? Currently I am doing the equivalent of using only the "I" signal. What would be the difference in terms of frequency analysis if any ?

thanks!
The focus is on what would be the difference between DFT output on just the I-data vs DFT output of I /Q- data.
Even so, how would the phase shift of LPF affect DFT analysis output ?