If you are right and I think you are, you must act quickly and hope the HD lasts long enough to retrieve all you want.
There are other ways - but I would ................ Fit the new HD as a secondary drive....................Use a boot floppy or CD to boot the computer and ...... if possible partition the new HD so that there is one partition big enough to replace your original drive C: and a second one hopefully at least as large to keep an image file temporarily, and perhaps more data from the old HD.
Format the new HD appropriately
Use Norton Ghost or Drive Image to make a clone of your failing Drive C: on the second partition of your new HD.
Now at last......"restore" the clone to the First partition of your new HD....... If you make it this far you are nearly home!!!!!!!!
Now make the New HD the Primary HD and the old one the secondary.
The computer should now boot normally and the old HD may or may not be accessible depending on its state of health now.
You can delete the clone image on your second partition to save space, or keep it in case something goes pear-shaped soon.
It is possible that you amy need to play with the BIOS early on, but hopefully any HD changes will be recognised automatically