I wound up solving this myself.
For whatever reason in my case App Engine stopped accepting the auto-verification of domain ownership from Google Domains and wouldn't recognize me as the owner of those domains.
In App Engine settings, go to the custom domains tab and click the "Add a custom domain" button. On the following page, select verify domain ownership from the dropdown and select the specific domain that you're locked out of (in my case it was
api.mydomain.com). This will take you to Webmaster Central* where you will need to click on the specific domain. Ignore the message that says you are the verified owner of the domain and click "Verify using a different method." On the following page I selected Google Domains from the list of providers and got a TXT verification code to copy.
I opened Google Domains in another browser tab. In order to add the TXT to my DNS configuration, I had to delete the associated CNAME records first (
api.mydomain.com). Once you have added the TXT record, go back to the Webmaster Central tab and hit the verify button. It will probably take > 5-10 minutes for it to pick up the TXT record. I left and came back. Once it verifies you as the owner, go back to App Engine settings > Custom domains tab and the greyed out trash can next to your domain should be clickable again. Delete the mapping. Once the mapping is gone you can go back and delete the TXT record in Google Domains.
*Note: I think it might be important to come to Webmaster Central via GAE instead of going directly to the site in another browser window. I could be misremembering, but when I went to Webmaster Cental directly and reverified my domain I did not regain access in GAE.
Hope this helps somebody.