Solved!
echo mypassword | sudo -u game-servers -S screen -dmS blockland0
echo mypassword | sudo -u game-servers -S screen -S blockland0 -X "GAMEMODE=\"trench_ctf\"; xvfb-run --auto-servernum --server-num=1 xterm -e 'wine ~/servers/blockland/server0/Blockland.exe ptlaaxobimwroe -dedicated -gamemode $GAMEMODE'"
It works now. I still need to configure sudo properly so I don't have to echo passwords.