Have Metz on the Thunderbird ( OEM ) whilst I don't have any problems with them I will change to the Avons or Michelins when they need it. As others have suggested tyre pressure, balancing them and wheel alignment are where I would look. If they have done a fair few miles which seem likely if the bikes done 13K you are most likely to be on a second set. From reading other forums they seem to get around the 8k miles from a front ( for your bike. ) less on the rear. Use a thread depth measure to see what the thread depth around the tyre is, I may give a better indication of what the profile of the tyre is like. Also again from other forums the Metz seem to be prone to tram lining a change of tyres to Avons or the Michelins cured this usually.
My front Metzler has 8500 miles on it, still looks good, wish it was worn out, but will be going soon. If I wasn't a tight Scottish git, I would have swapped at the same time as the rear Rear was squared off and tramlined horribly by 5500 miles, change to Avon cured this.
Heard a story a few months ago on a different forum where a guy was having all kinds of trouble with his bikes rear end sliding all over the place. He went through all the advice he got but in the end it turned out that he'd got some kind of resin coating on his back tyre. It was hardly visible and the tyres were not new so he'd picked it up while out somewhere. In the end he used a bit of sandpaper to rough up the tyre a bit and then pick all the hardened resin type stuff off. He never found out what it was or where it came from. Just thought I'd mention this in case of the 1:1,000,000,000,000 chance you're affected in the same way.
I'd put money on the wheels being out of alignment. I've had a similar problem with my CRF dual sport. The OEM knobbly tyres (IRC) were pretty crap but lasted pretty well for knobblies, probably because they were so hard, but the bike tracked true. I swapped them out for Bridgestone trailwings and the front felt like it wanted to tuck under in turns until I twiddled around with the alignment.
Whoa! Pasties and sausage rolls? You mad? You missed out on my Harley that actually spoke "potato potato" when the engine was running! Sounds a meal to me now!
Head stem bearings!!! Not heard a mention, they come loose sometimes. I've put tapered roller bearings in my head stem and now steers much better, my symptoms were steering was very erratic and there was a shudder and knock under heavier braking, anything sound familiar Dazza ? I've found in 30 plus years of riding that the tail wags the dog and the dog wags the tail, so what seems to be from the back originates in the front and vice-versa, so it pays to check both ends when you have handling issues.
To be honest, Life has got in the way of me getting near this as yet. I think I'll have a look at it next weekend. Its a pain to do without a center stand though.
To be honest, its sat in the garage at the moment. I dont like riding it in the dry, let alone in the wet/cold. I will get to it sometime over the winter and have a look at all of the aspects mentioned. I will keep you posted.