  The $2d$ Hubbard model with nearest-neighbour hopping on the square lattice
and an average of one electron per site is known to undergo an extended
crossover from metallic to insulating behavior driven by proliferating
antiferromagnetic correlations. We study signatures of this crossover in spin
and charge correlation functions and present results obtained with controlled
accuracy using diagrammatic Monte Carlo in the range of parameters amenable to
experimental verification with ultracold atoms in optical lattices. The
qualitative changes in charge and spin correlations associated with the
crossover are observed at well-separated temperature scales, which encase the
intermediary regime of non-Fermi-liquid character, where local magnetic moments
are formed and non-local fluctuations in both channels are essential.
