Ubuntu 14.04 MySQL 5.6 Upgrade Issue: Can't Start MySQL

I have a droplet that I had installed MySQL 5.5 upon initial setup of server a year or so ago but haven’t needed MySQL until recently. First, I wanted to update to 5.6 as it was what the project called for. So I ssh’d into the server and made sure everything was in working order first; logged into mysql, looked at the default test db etc, all was well. So I went on to updating:

sudo apt-get update
sudo apt-get upgrade
sudo apt-get install mysql-server-5.6

Everything appeared to be going fine, but upon exiting the install I get the following:

2017-04-12 15:00:51 0 [Warning] Using unique option prefix key_buffer instead of key_buffer_size is deprecated and will be removed in a future release. Please use the full name instead.
2017-04-12 15:00:51 0 [Warning] TIMESTAMP with implicit DEFAULT value is deprecated. Please use --explicit_defaults_for_timestamp server option (see documentation for more details).
2017-04-12 15:00:51 0 [Note] /usr/sbin/mysqld (mysqld 5.6.33-0ubuntu0.14.04.1) starting as process 6673 ...
start: Job failed to start
invoke-rc.d: initscript mysql, action "start" failed.
dpkg: error processing package mysql-server-5.6 (--configure):
 subprocess installed post-installation script returned error exit status 1
E: Sub-process /usr/bin/dpkg returned an error code (1)

And when trying to connect to mysql (mysql -u root -p) I get:

ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (111)

I’ve completely uninstalled, reinstalled, cleaned, autocleaned, and am at a loss. I even tried uninstalling 5.6 and trying to go back to 5.5 but that results in the same as above. What have I done? And what can I do to resolve it?

Thanks!