Recipe for Ubuntu 20.04:
Remove all shortcuts for language switching in Tweeks (Additional Layout Options -> Switching to another layout).
Just in case backup the parameters:
gsettings get org.gnome.desktop.wm.keybindings switch-input-source
# ['<Super>space', 'XF86Keyboard']
gsettings get org.gnome.desktop.wm.keybindings switch-input-source-backward
# ['<Shift><Super>space', '<Shift>XF86Keyboard']
Set the parameters:
gsettings set org.gnome.desktop.wm.keybindings switch-input-source-backward "['<Alt>Shift_L', '<Shift>XF86Keyboard']"
gsettings set org.gnome.desktop.wm.keybindings switch-input-source "['<Alt>Shift_L', 'XF86Keyboard']"
Check Settings -> Keyboard Shortcuts, filter by source keyword.
Switching to next input source should become Alt+Shift L.
Switching to previous input source should become Alt+Shift L.
Like on the picture below:

Switching language now should work by Alt + Left Shift. Holding Alt and pressing Left Shift should show standard language menu in the center of screen.
Ctrl + Alt + Shift + T shortcut should work by means of either left or right keys.
To be able to use, for example, Alt + Shift + C shortcut in IDE Shift should be pressed and hold first, then Alt + C.
Otherwise Alt first + Shift will switch the language.