I have 2 hard disks. Is it possible to span my /home directory over what's left of /dev/sda and the whole of /dev/sdb?I assume that I need lvm to do this. I don’t mind doing a fresh install, should it come to that. Thanks!

AFAIK, we don't offer to do this for you automatically. I suggest you set up lvm while keeping your current /home in place. Then copy your /home to the new lvm partition you created and add an entry for it in fstab at a location such as /lvm-home and reboot. If that all works then edit your fstab again to mount the lvm partition at /home.