Well, I’m not talking about getting drunk, certainly not on Listerine. The vigorously part I’ve found essential to get the alcohol in direct contact with the nerve, and also to wash away anything that might be pressuring the nerve.
Once I found blessed relief from a horrible toothache by just swirling hot water around my mouth for a couple of minutes–suddenly the filling on top of the cavity pops out, and no more pressure, no more pain.