How to disable the "tap dragging"-feature in Ubuntu 18.04 (libinput) without disabling "tap to click"?

GNOME has a setting for this, it is just not accessible in GNOME Control Center.

You can either use terminal or dconf-editor to access this setting.

Terminal:

1. Turn off tap-and-drag.

   gsettings set org.gnome.desktop.peripherals.touchpad tap-and-drag false
   

2. Check if it was successful.

   gsettings get org.gnome.desktop.peripherals.touchpad tap-and-drag
   

dconf-editor:

1. Install dconf-editor
2. Inside dconf-editor navigate to org/gnome/desktop/peripherals/touchpad
3. Find tap-and-drag and turn off the setting

Since 18.04 is using Xorg instead of Wayland, you should be able to set it this way:

First find the device:

% xinput list                                                                   
⎡ Virtual core pointer                      id=2    [master pointer  (3)]
⎜   ↳ Virtual core XTEST pointer                id=4    [slave  pointer  (2)]
⎜   ↳ SynPS/2 Synaptics TouchPad                id=12   [slave  pointer  (2)]
⎜   ↳ TPPS/2 IBM TrackPoint                     id=13   [slave  pointer  (2)]
⎣ Virtual core keyboard                     id=3    [master keyboard (2)]
    ↳ Virtual core XTEST keyboard               id=5    [slave  keyboard (3)]
    ↳ Power Button                              id=6    [slave  keyboard (3)]
    ↳ Video Bus                                 id=7    [slave  keyboard (3)]
    ↳ Video Bus                                 id=8    [slave  keyboard (3)]
    ↳ Sleep Button                              id=9    [slave  keyboard (3)]
    ↳ Integrated Camera: Integrated C           id=10   [slave  keyboard (3)]
    ↳ AT Translated Set 2 keyboard              id=11   [slave  keyboard (3)]
    ↳ ThinkPad Extra Buttons                    id=14   [slave  keyboard (3)]

I'm on a thinkpad. Your devices are likely different.

List the properties of the touchpad:

% xinput list-props 12                                                          
Device 'SynPS/2 Synaptics TouchPad':
    Device Enabled (140):   1
    Coordinate Transformation Matrix (142): 1.000000, 0.000000, 0.000000, 0.000000, 1.000000, 0.000000, 0.000000, 0.000000, 1.000000
    libinput Tapping Enabled (276): 1
    libinput Tapping Enabled Default (277): 0
    libinput Tapping Drag Enabled (278):    1
    ...

Then disable it:

% xinput set-prop 'SynPS/2 Synaptics TouchPad' 'libinput Tapping Drag Enabled' 0

The way I set this permanently is by putting into a script that runs on login.

I hope that helps. I don't know how to fix it at the system-level, nor on Wayland.