OK, we figured it out.  We had to take the "negative" approach rather than the "postitive" one to do it.  So in the first Variable Set, we would need to set up the Catalog Client Script like this:

Catalog Conditions: Field A2 is NOT "Other"

Reverse if False: Checked

Catalog UI Policies Actions:

Variable Name: Field A3

Mandatory: False

Visible: False

Clear the variable value: Checked

And do likewise for the other Variable Set.

When set-up like this, it works as intended.