Yes, but more importantly no.
The electrons, as you say, are wiggling back and forth as the voltage sweeps between positive and negative. But we’re not really interested in the motion of *the individual electrons,* but rather the propagation *of the electromagnetic field they interact with.*
The first quantity is called the “drift velocity,” and at one amp through a wire 2mm in diameter, this drift velocity is on the order of 8 centimeters per hour.
However, the second quantity, “signal velocity,” is what we’re really interested in when it comes to electric circuits, and that value is much closer to the speed of light in a vacuum.
https://www.wtamu.edu/~cbaird/sq/2014/02/19/what-is-the-speed-of-electricity/
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