Looking for tunes with vocals that are mostly spoken word yet still musical (e.g. The Distance by Cake).

Looking for good songs or artists in which the vocals aren't so much sung, but more spoken.

Not really rap, that's a different skill set. Not really spoken word poetry either.

Cake is the first example that comes to mind for me with songs like The Distance and Short Skirt Long Jacket.

He doesn't so much sing but it's still musically interesting and gels well the the rhythm and melody.