create a new screen session:
screen -AmdS namegoeshere
list all current screen sessions:
screen -ls
attach a running screen session:
screen -x namegoeshere
or
screen -x numbergoeshere (see screen -ls)
detach from a screen session:
press ctrl+a, then ctrl+d
Once you've created a new screen session, attach to it, start your server in there instead. Detach (or close the putty window) and it should keep running.