It seems you put some menu items behind what I'll call "[space] mode," where you you have to press the spacebar first to open the command menu, then use the command.
This is not reflected properly in the help text shown when you press ? and that was a source of confusion for me.
Especially since I managed to activate the fullscreen mode for one pane AND turn it off, but then couldn't figure out how I did it; and also, I did not find the space-Q option to Quit at first.)
Edit to add, I prefer installing with pipx.
These commands worked for me, to get Postgresql and MariaDB database plugins:
pipx install sqlit-tui
cd ~/.local/pipx/venvs/sqlit-tui
source bin/activate
bin/python3 -m pip install psycopg2
sudo apt-get install -y libmariadb-dev # On Debian
bin/python3 -m pip install mariadb
I didn't try installing system-wide as per the GitHub instructions, I don't know if that would have worked just as well with pipx or not.