Is schizoaffective disorder when you think other people are talking about you or saying things under their breath about you? but no hallucinations

No...maybe paranoia?: Look up schizoaffective disorder in the dsm-v (diagnostic and statistical manual of the american psychiatric association) as it give a a more complete explanation/description... But briefly, that patient has a mixture of mood disorder, usually depression, with some components of thought disorder, which could be anything from hallucinations, delusions or simply incoherence.