Here's one workaround without using javascript - just more embedded data, and branches in your survey flow.
Say

lastname

is the field with embedded data for your respondent's last name. We can create another embedded data field with the text that you want to appear in your question... we'll call it

lastname_qtext

here.
Add the embedded data for

lastname

in your survey flow. Beneath that, create a branch in your survey flow that respondents will go through with the condition that

lastname

is empty. Within that branch, create a new embedded data field called

lastname_qtext

, and set it to be whatever text you want displayed in the question for respondents missing last names. So it could be something like "DON'T" in this case.
If you want, you can create another branch for the condition that

lastname

is NOT empty.
When writing your question, use piped text for the new embedded data field

lastname_qtext

to create a question that will dynamically change based on the value of

lastname_qtext

(which was determined by the value of

lastname

). You'd just have to make sure that the question comes AFTER the branch in your survey flow where you define

lastname_qtext

.
Here's a screenshot of what your survey flow and question text could look like.

Here's what the question text could look like, with the piped text.
Here's what the question would look like when the last name is missing:
And here's what the question text would look like when the last name is present:

Hope that works for you!

---

Qualtrics.SurveyEngine.addOnReady(function () {
qc = this.getQuestionContainer().querySelector(".QuestionText");
lastname = "${e://Field/LAST_NAME}";
if (lastname == "") {
qc.innerText = "We DONT HAVE your last name";
} else {
qc.innerText = "We have your last name";
}
});

---

I didn't just get one working answer, but two! Tested both solutions, and they both work. Thanks!

---