Hello, @Jessica M. McGeary,
Welcome to the Microsoft Q&A platform!
Based on your description, I understand that you do not want users to download attachments from your draft emails on other devices. Unfortunately there is no built-in functionality in Exchange to fulfill your needs.
I know that there is the ability to set up conditional access policies based on device in Microsoft Entra, which is similar to your needs, and you can check out the following document to get a general idea: https://learn.microsoft.com/en-us/entra/identity/conditional-access/concept-condition-filters-for-devices?source=recommendations.
If the answer is helpful please click on ACCEPT ANSWER as it could help other members of the Microsoft Q&A community who have similar questions and are looking for solutions.
Best Wishes,
Alex Zhang