It's a common mistake that the warp core is the primary power plant for the whole ship. That'd be wasteful, as they use fusion plants to make antimatter in the first place.

TNG tech manual states that starbase antimatter production is around 25% efficient. Starships can make their own, but it implies that it's even less efficient than a starbase.

Most of the ship gets power from fusion. Those can't dump enough energy at once to power higher warp factors (limit is maybe warp 3), though, so you use energy while not at warp to make antimatter, which can then be dumped into the warp core later.

Shields will work fine with just the fusion plants.