Hi Grover,
You'll find lots of discussion of this if you do some searching in the archives.
An easy first thing to check is which hole the clutch slave cylinder push rod is connected to on the clutch release shaft end lever. There are three holes drilled in these. The center one should be used on all TRs.
Another easy thing to check is the pedals themselves, just disconnect at the master cylinders and operate the pedals by hand to make sure they are smooth and not binding anywhere.
Yet another easy thing to check is bleed the hydraulics, to be sure there is no air in the system, and change the fluid if you think it's old.
But, most likely, you have a broken release fork taper pin. The replacements provided by the vendors for many years were sub-standard and sheer off easily. Even the originals weren't all that strong and could break. If the fork is rotating a few degrees on the shaft, due to the fixing pin sheered off, it will cause symptoms like you describe.
The fix is to remove the broken pin, replace it with a new, hardened type, and many wise TR owners back this up with a secondary fastener installed through the clutch release fork and shaft. This job means pulling the gearbox out of the car, and a full clutch job is usually recommended while it's all apart.
Another possibility is the bushings the clutch release shaft rides in, pressed into the bellhousing, which might have worn badly. Later gearboxes were fitted with 3/8", non-greaseable bushings. Earlier gearboxes used 3/4" greasable fittings that are much more durable if given proper service. The grease fittings are on the ends of the shaft, and the bearings are drilled to allow grease to squeeze inside. If 3/4" bushings can't be found, it's possible to install two 3/8" per side, or four 3/8" total, for an improvement. The greaseable shaft is otherwise the same and fits either tyle of bushing.
It's possible the hydraulics need attention, either slave or master cylinder. It sounds like you don't drive the car a lot (low mileage) and that can lead to gumming up in there.
The TR6 clutch arrangement is prone to problems. One reason is it uses a fixed, non-adjustable push rod at the slave cylinder. The earlier cars use an adjustable push rod that makes it possible to keep the clutch release nicely tuned. I think Triumph was looking to make the setup require little or no maintenance, but shortened the life of the clutches, as a result. However, if I had a TR6 I'd replace the push rod with an adjustable type. I've seen these available on eBay, but the earlier type looks like it should work and is widely available from vendors.
There are several different brands of replacement clutch pressure plates and driven disks, too. Sometimes these have stronger springs than original, that might lead to early throw out bearing failure. Some folks go to a Gunst bearing (TRF sells them, description on their website) as a solution.
Installation of the original type of throw out bearing makes a difference too. There is a special procedure that needs to be followed. If not, the bearing will usually fail early.
However, throw out bearings usually get noisy when failing, and I suspect from your description it might be something else. Still, it's possible.
Finally, apparantly the 6-cylinder cars, or at least some of them, don't have permanently fixed dowel pins in the engine flange to gearbox bellhousing. (Earlier 4-cylinder TRs did.) Knowledgeable installers put dowels in, either permanently or temporarily, to better align the gearbox and engine. If this isn't done, slight misalignment can lead to early clutch failure.
Any of the above can lead to clutch not disengaging or engaging fully, and early failure of the driven plate. I'm afraid you might be looking at a full clutch job, and some investigation into the root cause or causes of the problem. That done and all problems addressed, you should get ten times the mileage you have out of this current clutch.
Alan