I finally managed to get Elasticsearch running. The problem was that the uninstallation/upgrade process had left orphaned files from version 6.0.1. The solution was to completely delete /usr/share/elasticsearch/* as mentioned here: Version 6.4.2 does not start (java.lang.NoSuchFieldError: LOG_DEFAULT_LEVEL_SETTING) and make sure Elasticsearch 6.0.1 was completed uninstalled before installing a fresh copy of 7.6.2.
Additionally, I had to chown -R elasticsearch:elasticsearch /opt/elasticsearch/ (the location of my indices) because the old elasticsearch user was still set as the owner of the directory and the fresh install of 7.6.2 had not corrected that.
I also encountered the same problem mentioned here Problems with access to Elasticsearch form outside machine and setting discovery.seed_hosts: [] solved that.