What happens in the synapse when cocaine administration occurs in the human brain
You are correct that cocaine "fills the tunnel" (the pharmacology terminology is that it "blocks the transporter").  But it is not an autoreceptor, so the reuptake transporter does not send a stop signal, it just removed neurotransmitter from the cleft.  But once you've blocked it with cocaine, neurotransmitter is not being removed from the cleft so it has a much longer and stronger signaling effect on the post-synaptic neuron.

But cocaine can also block sodium channels, which assist in action potential propagation and that has more of a local anesthetic effect.

The effect you are thinking of would pertain to auto-receptors, not reuptake transporters.  Autoreceptors tend to be more like the post-synaptic targets of the neurotransmitter.