As mentioned by Pat in the comments, my home and siteurl config values were set to https://www.mysite.com and https://app.mysite.com respectively. These values are held inside your site’s database. I think you can change them through the WordPress admin page, or you can enter your database directly and find them with the following command:
SELECT * FROM wp_options WHERE option_name="siteurl" OR option_name="home";
In my case, the protocol for siteurl and home was https not http. I believe this meant WordPress was redirecting traffic to https://www.mysite.com, but as my virtual hosts file doesn’t handle that traffic this caused a redirect to a public DNS server and gave me the incorrect site.
(Could someone with more experience please confirm if this accurate?)
To fix the issue, I altered the siteurl and the home options to use the http protocol, which can be done from within your database using the following commands:
UPDATE wp_options SET option_value="http://www.mysite.com" WHERE option_name="home";
UPDATE wp_options SET option_value="http://www.mysite.com" WHERE option_name="siteurl";
I used these commands on the databases for both my sites, and it resolved the problem.