Copy and Paste.
You may need to enable the new drive in DISK MANAGEMENT or else you may not see it. My recommendation:
1) Unhook current drive for now to avoid further damage (simply unhook both cables from rear of drive when PC is off)
2) Attach new drive, then use Disk Management if needed
3) Run a FULL FORMAT (overnight) to build up the bad sector table on the new drive
4) Attach the old drive
5) COPY and PASTE the folders you need
6) Shutdown, remove drive and send back as directed.