See the GNU Screen chdir command. All new windows created in Screen use this as their initial directory. Using this, you can do something like:
chdir /home/dan/newscreendir
screen
And your new window (along with any future created windows) will be in the set directory. If it’s always going to be the current working directory you may be able to set something up in your screenrc to do this one in one command.
See the GNU Screen man page. It’s quite comprehensive.
Screen chdir command
Screen cannot access your shell variable nor execute backticked commands. The closest I can get to doing it in one click is with a small Bash script like this:
screen -X setenv currentdir `pwd`
screen -X eval 'chdir $currentdir' screen
Or more compactly:
screen -X eval "chdir $PWD"
screen -X sends the command to the currently running Screen session. The first line creates a variable called currentdir. The second line sends the currentdir to the chdir command and then creates a new Screen window.