But when I copy and paste and try to move this 5/8" Diameter hole, it continues the dimension line horizontally, and does not simply replicate another 5/8" Diameter hole:

Individually dimensioning each circle is challenging because I cannot “Snap to” the full diameter of the circle, as the circles do not correspond to grid lines.

Anyone know how I can multiply these dimensions across all of my circles?

As you didn’t post the file, I can’t be sure, but I suspect one end of the original dim is snapped to a grid point and not to the circle, hence the extension when you place a copy elsewhere.

But to the point. To snap your dimensions to the circles, tick Quad snaps and turn on Osnap (and turn off Grid Snap). There are four quads to a circle at N, S, E and W and either opposing pair will give you a diameter.

And then: why not use the Diameter dimension (if you’d rather type commands than pick from a menu, this is _DimDiameter) which is the usual way to dimension a circle?