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The fuzzy dark matter (FDM) scenario has received increased attention in recent years due to the small-scale challenges of the vanilla Lambda cold dark matter ($Λ$CDM) cosmological model and the lack of any experimental evidence for any candidate particle. In this study, we use cosmological $N$-body simulations to investigate high-redshift dark matter halos and their responsiveness to an FDM-like power spectrum cutoff on small scales in the primordial density perturbations. We study halo density profiles, shapes and alignments in FDM-like cosmologies (the latter two for the first time) by providing fits and quantifying departures from $Λ$CDM as a function of the particle mass $m$. Compared to $Λ$CDM, the concentrations of FDM-like halos are lower, peaking at an $m$-dependent halo mass and thus breaking the approximate universality of density profiles in $Λ$CDM. The intermediate-to-major and minor-to-major shape parameter profiles are monotonically increasing with ellipsoidal radius in $N$-body simulations of $Λ$CDM. In FDM-like cosmologies, the monotonicity is broken, halos are more elongated around the virial radius than their $Λ$CDM counterparts and less elongated closer to the center. Finally, intrinsic alignment correlations, stemming from the deformation of initially spherically collapsing halos in an ambient gravitational tidal field, become stronger with decreasing $m$. At $z\sim 4$, we find a $6.4 σ$-significance in the fractional differences between the isotropised linear alignment magnitudes $D_{\text{iso}}$ in the $m=10^{-22}$ eV model and $Λ$CDM. Such FDM-like imprints on the internal properties of virialised halos are expected to be strikingly visible in the high-$z$ Universe.