Is it possible for you to add the 1TB drive as a second drive? Thereafter as other members have suggested you can use CloneZilla to clone those partitions that are running out of space onto the 1TB drive.
Are you using LVM at all? If you are, then you could add the 1TB drive as a new PV; and grow the relevant Logical Volume.
Perhaps post the output of:
In this way we'll be able to advise you more.