+1  A: 

jtolle steered me in the right direction. As far as I can tell, this is what I was trying to do. I've been testing and it seems to work.

stringOriginFormula = rangeOrigin.Formula
rangeOrigin.Cut rangeDestination
rangeOrigin.Formula = stringOriginFormula

Thanks jtolle!

Kuyenda
You're welcome. A nice example of reframing the problem leading to a (much) easier solution. You might want to edit the question title...maybe "How can I get literal references in formulas to update when they are moved?" or something? Technically that's now "not programming related", but I think our discussion in the comments justifies keeping the question around.
jtolle
Those are lines of code from my procedure. I am not doing this with a mouse, so we are safely in the programming related zone.
Kuyenda