well i dont quite get what you're trying to ask but definitetly we cant get the desired signal if the locally generated carrier is still at that particular frequency and the incoming signal starts to shape away and vice versa.i can only tell about DSB coz i've only completed the study of DSB.SC YET.
m(t)cos(ω°+Δω)t multiplied by cos(ω°t) will yield after passing through LOW PASS FILTER m(t)cos(Δω)t and not m(t)
The above said explaination is right the effect of this would even make the outhput of the detector to zero, moreover phase change can also occur. This is obvious bcoz we cannot make both cosine term exactly equal in phase due to this the message is multiplied by this phase difference
i think it is obvious, the same effect will result even in SSB, anyway we are using the same type of receiver( costos receiver i.e synchronous detection)