You do not need to apply any special logic for the expiration of an access token near the boundary of the expiration time; the Dropbox API servers will authenticate each request and either allow or reject it. For any given API request, the request will either be authenticated and then proceed, or it will not (e.g., if the access token is expired) and the call will be rejected. The call will be authenticated at the beginning of the request, and then the file upload will proceed. As long as you receive a successful API response, then the file was uploaded. If you receive an API error, then the file was not uploaded.
If the call is rejected, indicating that the short-lived access token is expired, you should then get a short-lived new access token for further calls. You can do so by having the user re-authorize the app, or if your app requested "offline" access, use the refresh token to get a new short-lived access token programmatically. You can find more information in the OAuth Guide and authorization documentation.