Can you tell me a bit more about how your referral codes are to be generated? Additionally, how are you currently planning on tracking the referral codes?

If I'm thinking about your issue correctly, I would recommend creating two referral code for each participant. For example-- "123A" and "123B".
If you know these referral codes ahead of time, then you can upload them into a Qualtrics contact list and use the code authenticate participants upon entry to your survey.
Note that this all assumes you'll know these referral codes ahead of time. Is this the case? Otherwise, how are referral codes generated?
*STEPS:*
Create two columns in a CSV file with the following headers- "Email" and "ExternalDataReference". You can keep the Email column blank, but it's required to upload anything into a Qualtrics contact list; put your list of referral codes underneath the "ExternalDataReference" column.
Next, upload your CSV file with the codes into a Qualtrics contact list.
Next, in your survey flow, add an "authenticator" element (https://www.qualtrics.com/support/survey-platform/survey-module/survey-flow/advanced-elements/authenticator/authenticator-overview/), and use the contact list you just uploaded. It might look something like this within your survey flow:
Participants will be asked to enter their referral code upon opening the survey. Once someone has used a given authenticator code to complete the survey, that code cannot be used again.

Thanks so much for your thoughtful answer. We can generate codes ahead of time if needed, but either way the plan is to only have one unique code per referrer to minimize the burden on them. They would share this same code with up to 10 of their eligible contacts, and the code would cease to work after it's been used to complete 2 surveys.
The strategy you mention sounds like it would need each referrer to send a different code to each individual referee, right? (I don't think 2 codes would work for up to 10 referees.) That's interesting to consider, but I think placing the responsibility of keeping track of all 10 codes so as to avoid double-sharing them might be too much to ask (of thousands of participants).
Even then, as far as I can tell there's nothing making sure no more than 2 codes from any given set were used to finish a survey, which is the main objective we have here.