IMac hard drive failing; external TM will restore but won't then back up

It appears my early 2009 (OS X 10.9.5) hard drive is failing as it crashed 6 months ago, in November and again this week. SMART Utility says it has re-allocated 6 bad sectors and there has been 1 re-allocated event. Each Time I have successfully restored from an external hard drive with TM on it. Prior to the November fail, CCC reported an odd (unknown to me system file) corrupt file but, instead of replacing the file as perhaps I should, I decided to re-boot which it wouldn't do and hence the November TM restore.
However, despite restoring successfully the external TM will no longer make back ups. I have tried repairing it but it is greyed out and Verify says 'unable to complete.' Eventually I did get the greyed out button to work but Repair reported:
Verifying volume “Time Machine”
Checking file system
Checking Journaled HFS Plus volume.
Checking extents overflow file.
Checking catalog file.
Keys out of order
Invalid extent entry
Incorrect block count for file iNode785919y8
(It should be 13 instead of 4194317)Invalid extent entry
Missing thread record (id = 82)
Incorrect size for file iNode78591999
(It should be 4096 instead of 274877907467)
Invalid extent entry
File record has hard link chain flag (id = 78591999)
File has incorrect number of links (id = 78591999)
(It should be 1 instead of 17)
Missing thread record (id = 18)
Invalid extent entry
Invalid extent entry
Invalid extent entry
Incorrect block count for file iNo$e78592002
(It should be 4194317 instead of 13)
Missing thread record (id = 82)
Invalid extent entry
Incorrect size for file iN/de7x592003
(It should be 4096 instead of 274877907467)
Invalid extent entry
File record has hard link chain flag (id = 78592067)
File has incorrect number of links (id = 78592067)
(It should be 1 instead of 17)
Missing thread record (id = 4194322)
Invalid extent entry
Invalid extent entry
File record has hard link chain flag (id = 78592004)
File has incorrect number of links (id = 78592004)
(It should be 1 instead of 17)
Missing thread record (id = 18)
Incorrect size for file iNode78592005
(It should be 0 instead of 64)I
nvalid extent entry
Invalid extent entryIncorrect size for file iNode78592006
(It should be 0 instead of 4194304)
Invalid extent entry
Invalid extent entry
Missing thread record (id = 4194322)
Invalid extent entry
Invalid extent entry
File record has hard link chain flag (id = 82786312)
File has incorrect number of links (id = 82786312)
(It should be 1 instead of 17)
Missing thread record (id = 18)
Incorrect block count for file )Nod%78592009
(It should be 65 instead of 1)
Invalid extent entry
Invalid extent entry
Incorrect block count for file iNode78592010
(It should be 0 instead of 4194304)
Invalid extent entry
Incorrect size for file iNode78592011
(It should be 0 instead of 274877906944)
Invalid catalog record type
The volume Time Machine could not be verified completely.
Error: This disk needs to be repaired. Click Repair Disk.
Whilst pondering this, the hard drive crashed again this week. Bizarrely I was able to restore from the very same external TM that will not let me do further back ups nor be repaired.
I wonder if there are any further checks I can do to establish if my iMac hard drive is failing? I will then decide whether to replace the hard drive - I see I can fit a WD 6TB with the jumpers set to reduce the speed to 3GB/s - or buy a new iMac.
I wonder also whether there is anything I can do to preserve my TM back ups. I realise I am being guided to re-format this external drive, but can I do that and then add back the actual existing back ups (which I would transfer to another disk before re-formatting) rather than start again from no history newly formatted?

Omdineen wrote:
It boots up to where its just my background and the Finder bar but then crashes, so I can't click anything. Anything from here or am I just screwed?
There are a few remaining options.
If you have another Mac with a FireWire port, try FireWire Target Disk Mode: http://support.apple.com/kb/HT1661
You need a FireWire cable to connect the two Macs. This will allow you to mount your failing HD on a working Mac and copy files that way.
Try Safe Mode - read about it first:
Safe Mode or "Safe Boot" is a troubleshooting mode that bypasses all third party system extensions and loads only required system components.
Starting up in Safe Mode 
This assumes the Finder is crashing due to a third party system extension or optional system component that Safe Mode will prevent from loading. You should still be able to copy files directly to an external device, but file sharing will be disabled.
Lastly boot from the DVD again, repair the disk again, reinstall OS X, and keep your fingers crossed. You're squeezing the last drops of life from the thing.

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