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Fully 3D cosmological simulations of scalar field dark matter with self-interactions, also known as Bose-Einstein condensate dark matter, are performed using a set of effective hydrodynamic equations. These are derived from the non-linear Schrödinger equation by performing a smoothing operation over scales larger than the de Broglie wavelength, but smaller than the self-interaction Jeans' length. The dynamics on the de Broglie scale become an effective thermal energy in the hydrodynamic approximation, which is assumed to be subdominant in the initial conditions, but become important as structures collapse and the fluid is shock-heated. The halos that form have Navarro-Frenk-White envelopes, while the centers become cored due to the fluid pressures (thermal + self-interaction). The core radii are mostly determined by the self-interaction Jeans' length, even though the effective thermal energy eventually dominates over the self-interaction energy everywhere, a result that is insensitive to the initial smallness of the thermal energy. Scaling relations for the simulated population of halos are compared with Milky Way dwarf spheroidals and nearby galaxies, assuming a Burkert halo profile, and are found to not match, also for core radii $R_c\approx 1\text{kpc}$, which has generally been the value expected to resolve the cusp-core issue. However, the simulations have a limited volume, and therefore a limited halo mass range, include no baryonic physics, and use fiducial cold dark matter initial conditions with a cut-off near the Jeans' length at $z=50$, all of which can affect the halo properties. [Abridged]