I actually think I found a solution. I can do something like this, which will also allow me to copy and paste the formula to multiple rows, but when I cut data the cell references, the formula wont change. Example:

=INDIRECT("C" & ROW() & "");

When this is pasted into Column A, for example, and I drag it for 100 rows, it will always reference the C column in that row, even if I cut/paste data in the C column and move it elsewhere.

Thanks for letting me know about the INDIRECT() function though, that helped a bunch!