I have a '68 L-79 327 ci 350 HT with a 4 spd close ratio.  It is numbers matching original to the car.  I have spoken to all the owners going back to 1997 and nothing has been done to the engine or trans since then.

I would suggest changing the gear lube, and checking the clutch adjustment. Not likely that your trans is going out, just needs a little servicing. 

I am a little suspect that your clutch is not getting fully disengaged.  Check the linkage adjustment and be sure your floor mat is not getting in the way.  One way to check your clutch is try going into reverse when all is cold.  Here's what you do: Depress the clutch pedal all the way, then try 1st. gear.  If it doesn't want to go into 1st., try reverse.  If the trans grinds while trying to engage reverse gear, pull back to neutral, wait a couple of seconds and try reverse again.  If the trans still grinds, then for sure your clutch is in need of adjustment or replacement.

Per the owners manual included with the car
"To engage reverse, while at a complete stop, first put the car into 2nd or 3rd gear and then while still depressing the clutch pedal, shift into reverse, otherwise repeated use may damage the transmission".

The reason is your shift into a forward gear will momentarily stop the spinning secondary shaft holding the gears. It is spinning due to the input shaft spinning and the viscosity of the gear lube is causing the 2nd shaft to turn. This allows you to engage reverse without grinding. If first is also grinding, either yoiur clutch is out of adjustment or your 1st gear synchro mesh is wearing out. Usually 2nd is the first gear to lose its synchro since it gets more shifts than any other gear and first is typically engaged with the car standing still. A worn out clutch disc will improve the above situation rather than aggregate it since the disc gets thinner providing greater disengagement gaps.

Now, my last thought would be to as suggested above, try changing the lube. I'd bump up to a full synthetic that will flow at the lower temps. No viscosity thickening like the mineral based oils when the temps drop. It also offers improved cushioning when hot. Your shifts will be smoother while extending the life of your synchros and bearings. Good stuff!