Can I show an add-on only after the shopper picks a certain option?

Yes, with conditional logic. In Products → Add-ons, select the group and click Edit on the option that should appear only sometimes, for example the engraving text box.

1. Open Advanced — help text, cart labels, conditional logic, email.
2. Under Show this option when all / any of these match, click + rule.
3. Pick the controlling option in (field), choose an operator such as equals, and enter the value.
4. Click Update option.

With no rules the option is Always shown. Options with rules show a small "cond" tag in the list.