Your 300Tdi has a salisbury rear diff, and it is very unlikely that there is a problem with the diff itself unless it has been run without oil.  Replacing the diff centre only on the salisbury is not a trivial job, and this and the fact it is unlikely to be needed will be why they don't like that option.
As Mark2 says, the thing to look at that is likely to leave you stranded is a drive flange with stripped splines - these, and the axles themselves if necessary can be replaced in a few minutes without even jacking it up. Check whether this is the problem. 
Make sure the problem is not worn universal joints - failure of these can be disastrous. The most likely other thing is, as mark2 also points out, the ball joint on top of rear axle. While unlikely to strand you, it will only get worse. 
A dragging handbrake will make any transmission slack seem a lot worse than it is, so check that as well. Free play in any  of the rear axle locating bushes will show up as slack, but will mostly also cause steeringproblems that you will notice first. 
Unless you feel competent to diagnose exactly what the problem is yourself, get someone who is competent to do so and have them explain the problem and quote on fixing it. 
John
				
			 
			
		 
			
				
			
			
				John
JDNSW
1986 110 County 3.9 diesel
1970 2a 109 2.25 petrol
			
			
		 
	
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