In order to get around these issues, I first moved all of my private GitHub repos to Bitbucket. I didn't realize that Bitbucket was provided by Atlassian (same folks who do Jira, Stash, Confluence, etc.). I found it extremely easy to import my GitHub projects. They did a great job with that feature.
Then, onto moving the LDS Temples web service. For this, I decided to go with OpenShift since they allow 3 free "gears" indefinitely. This was a little more challenging, but since my app was already deployed as a WAR, it wasn't too bad. The only part that really bugged me is that they do not support apps written in Java 8 for their Tomcat 7 deployment feature. However, once I downgraded to Java 7, things worked smoothly after that. The webservice is now running here:
Anyway, that was my fun for the day. Now I need to update the app to use the new REST endpoint, since we no longer own the blinkdroid.com domain. It was costing too much.