Ubuntu – Issues with Unity Tweak Tool on Ubuntu 16.04

I recently installed 16.04 on my desktop, and one of the first things I did was install the Unity Tweak Tool (via the Ubuntu Software app store). It was working fine for a while. Later, after making some more customizations (installed Cairo-Dock, but it was performing terribly, so I removed it and replaced it with Docky), I eventually noticed that the Unity Tweak Tool icon on the dock was a gray box with a question mark, like it had been removed. I checked a terminal to see if I could just launch it via command line, but now the OS said it was not installed.

I've been attempting to reinstall the Unity Tweak Tool without any luck:

If I try clicking the install button in Ubuntu Software for the tool, it looks like its downloading for a moment, but then the Install button returns (yet, the unity launcher shows the icon and says "waiting to install"). Also, it doesn't even bother asking me for my password. I have no problem installing other apps with Ubuntu Software.

If I try to install unity-tweak-tool via apt-get, I get the following response:

Reading package lists... Done 
Building dependency tree 
Reading state information... Done 
Some packages could not be installed. 
This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:

The following packages have unmet dependencies:  unity-tweak-tool :
Depends: unity-webapps-common but it is not going to be installed 
E: Unable to correct problems, you have held broken packages.

Yet if I run dpkg --get-selections | grep hold there is no output, suggesting that there are no held packages. I have tried sudo apt-get clean and retried to no avail. Then I tried using aptitude instead, which gave some strange results too. It first said that 46 packages using 44.4 MB were going to be installed, and then it gave me a long list of unmet dependencies and its solution to meet them. I typed "Y" to agree and got the following response:

No packages will be installed, upgraded, or removed. 0 packages upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

Need to get 0 B of archives. After unpacking 0 B will be used.

I have also tried synaptic package manager. When I mark unity-tweak-tool for installation, it a) labels it as a broken package and b) shows a long list of packages that need to be removed (including a lot of Unity/Ubuntu stuff, and even synaptic itself, which makes no sense).

Is there any solution to this problem that does not require a fresh install of Ubuntu?

Also, my sources.list file looks like so:

#deb cdrom:[Ubuntu 16.04 LTS _Xenial Xerus_ - Release amd64 (20160420.1)]/ xenial main restricted

# See http://help.ubuntu.com/community/UpgradeNotes for how to upgrade to
# newer versions of the distribution.
deb http://us.archive.ubuntu.com/ubuntu/ xenial main restricted
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial main restricted

## Major bug fix updates produced after the final release of the
## distribution.
deb http://us.archive.ubuntu.com/ubuntu/ xenial-updates main restricted
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial-updates main restricted

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu
## team, and may not be under a free licence. Please satisfy yourself as to
## your rights to use the software. Also, please note that software in
## universe WILL NOT receive any review or updates from the Ubuntu security
## team.
deb http://us.archive.ubuntu.com/ubuntu/ xenial universe
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial universe
deb http://us.archive.ubuntu.com/ubuntu/ xenial-updates universe
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial-updates universe

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu 
## team, and may not be under a free licence. Please satisfy yourself as to 
## your rights to use the software. Also, please note that software in 
## multiverse WILL NOT receive any review or updates from the Ubuntu
## security team.
deb http://us.archive.ubuntu.com/ubuntu/ xenial multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial multiverse
deb http://us.archive.ubuntu.com/ubuntu/ xenial-updates multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial-updates multiverse

## N.B. software from this repository may not have been tested as
## extensively as that contained in the main release, although it includes
## newer versions of some applications which may provide useful features.
## Also, please note that software in backports WILL NOT receive any review
## or updates from the Ubuntu security team.
deb http://us.archive.ubuntu.com/ubuntu/ xenial-backports main restricted universe multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ xenial-backports main restricted universe multiverse

## Uncomment the following two lines to add software from Canonical's
## 'partner' repository.
## This software is not part of Ubuntu, but is offered by Canonical and the
## respective vendors as a service to Ubuntu users.
deb http://archive.canonical.com/ubuntu xenial partner
deb-src http://archive.canonical.com/ubuntu xenial partner

deb http://security.ubuntu.com/ubuntu xenial-security main restricted
deb-src http://security.ubuntu.com/ubuntu xenial-security main restricted
deb http://security.ubuntu.com/ubuntu xenial-security universe
deb-src http://security.ubuntu.com/ubuntu xenial-security universe
deb http://security.ubuntu.com/ubuntu xenial-security multiverse
deb-src http://security.ubuntu.com/ubuntu xenial-security multiverse