The flash head is a few inches from the center of the lens, and cannot be directly on lens axis, so there will always be some shadow. But if close to lens, then directly above it is the best place. Then the shadow is mostly hidden, directly behind the subject. Yes, there can be a shadow below the subject (hair, or arms, etc), but that's much less obnoxious than one at their side when camera is in portrait orientation. The purpose of the bracket is to keep the flash directly above the lens in either orientation. Having the subject stand a bit farther from the wall will help a little.
Off-camera flash is another subject, and is a strong plus, but at minimum, it needs to be high out at full arms length (on a hot shoe cord). Just a few inches away from lens on a bracket is flat light, and is only enough to cause the problems (unless directly above the lens). But the greater distance is better lighting (adds tonal gradients, not flat), and makes handling the shadow easier, but some shadow attention is always required (needs greater distance to wall, or further off axis, both of which places the shadow off the edge of the frame, etc).
Not your meaning, but to me, that sounds like either bounce, or an umbrella for the flash (instead of the bracket). Have you ever tried an umbrella? It's like magic.
A little diffuser on the flash cannot help at all. Because it is still only a couple inches of size, and still makes dark sharp shadows. Diffusers just spread the light, and all a tiny spreader can do is to spread it outward, to miss the subject entirely. But a 40 inch umbrella is large, and the outer edges on either side can spread the light inward to hit the subject. One ray path fills the other ray paths shadows. Then you really have something. The answer is a
large light.
See some samples (direct, diffuser, bounce, umbrella):
Creating Soft Light from the Flash and the next page too.