How to avoid overtones in audio signal
A good stringed instrument music pitch detection/estimation algorithm will do the opposite, e.g. it will not ignore overtones.  Instead it will pay attention to the harmonics, specifically the harmonic train and its spacing, as this is a stronger indication of human perceived pitch than spectral content at the fundamental frequency (which could be nearly or completely missing, depending on the microphone and channel characteristics).

If you want to work in the frequency domain, than look into cepstrum/cepstral analysis and/or the Harmonic Product Spectrum algorithm, which can estimate pitch from spectral periodicity.  Or you can instead work in the time domain using weighted/interpolated autocorrelation, ASDF, AMDF, etc. type algorithms for pitch estimation.

If you slice your time windows too small, then there will be very few full cycles at he fundamental pitch frequency, which causes FFT frequency estimation to have very poor resolution.  A parametric estimator within and across windows may work better than a bare FFT.