A few more days of tinkering and I got the code working. I needed a different way to test if a cell was blank (since it is blank, but not empty) I switched >0 for <>"" and now it works! Here is the working code:

=IF(Z3="","",IF(AND('Sheet2'!$B$3=FALSE,Z3>0),TRUE,IF(AND('Sheet2'!$B$3=TRUE,Z3>0,W3<>""),TRUE,FALSE)))