Ubuntu – Kernel did not upgrade between 14.04 LTS to 16.04 LTS

Yesterday I upgraded my Ubuntu Desktop from LTS 14.04 to LTS 16.04. I did it via Software Updater.

There were several errors during the upgrade (is there a place where these errors are listed in the system so I can post it here?). The last one of them was at the end, saying that the boot partition was full. Following a post, I deleted a few kernels which I thought were old.

Now when I do uname -r it tells me that the kernel version is 3.13.0-93-generic, but it should be 4.4 something right? Can it be that I deleted the correct kernel by mistake? If yes, how do I fix this?

But the command cat /etc/os-release shows:

NAME="Ubuntu"
VERSION="16.04.1 LTS (Xenial Xerus)"
ID=ubuntu
ID_LIKE=debian
PRETTY_NAME="Ubuntu 16.04.1 LTS"
VERSION_ID="16.04"
HOME_URL="http://www.ubuntu.com/"
SUPPORT_URL="http://help.ubuntu.com/"
BUG_REPORT_URL="http://bugs.launchpad.net/ubuntu/"
UBUNTU_CODENAME=xenial

The system seems to think it is 16.04 but I have a feeling there are many other wrong things, I just don't know how to find these upgrade errors.

Is there an official ubuntu post upgrade script that checks if everything is ok?

I don't know if it adds anything, but two other errors that I remember were that the ubuntu software center did not work after the upgrade, I fixed following a post that said it had something to do with a language setting. The other error was that on boot it showed "Lvmetad is not active yet, using direct activation during sysinit" which I fixed by edditing the GRUB file.

Anyways, I just want to know if my system is ok and what is the meaning of this contradiction between the kernel and the system version (if 16.04 had the wrong kernel would it even boot?)

Thank you very much in advance for any help!!!

Edit: Here is what I have installed (output of dpkg -l | grep linux-image):

linux-image-3.11.0-19-generic                                                               
linux-image-3.11.0-20-generic                                                               
linux-image-3.11.0-22-generic                                                              
linux-image-3.13.0-93-generic                                                              
linux-image-extra-3.11.0-19-generic                                                          
linux-image-extra-3.11.0-20-generic                                                          
linux-image-extra-3.11.0-22-generic                                                       
linux-image-extra-3.13.0-93-generic