For me, there are two channels of training / instruction if you like.
You've got the mental game of roadcraft; that game of chess with vision, planning, scanning, vanishing points ... and then you've got the physical game of how we control our bike; all the things that have been mentioned from counter steering to brakes and gears and body positioning. And those two things run pretty much hand in hand.
I still remember, like it was yesterday, my first 'big bike'. I was living in London, commuting from the isle of dogs, and I specifically remember really struggling to turn it on wet shiny bitumen roads, having just got it, in the middle of winter. (This was 1984 ...

)
One day, I picked up some book in WH smiths, 'your first super bike'

and I remember, eyes wide open, reading about counter steering! I never knew about it, jeez, I'd never even heard of it - and I remember feeling really cheated, that it had never even been mentioned when I'd done my training and taken my test.
And that had been my real sticking point - a wet shiny right hand bend, and me getting nearer and nearer the parked cars on my nearside as I tried to turn.
So I hear your pain, and I can almost see you struggling on those wet shiny roads, the odd sniff of diesel, tension in your arms, target fixating on the road surface, and really struggling to relax on the bike.
So, first off, check yer tyres. The difference between good and bad tyres on these winter roads, is chalk and cheese. Honestly - it really is. If you're running on something made of bakelite, bin them and get something decent. PR3's, Road smarts, TR91's ... that sort of thing. And this is even more relevant in my opinion, in town. Contrary to what some may think about wet country lanes, they generally have a damn site more grip than a wet urban ring-road. (And check your tyre pressures ...).
Knowledge is power. Carefully analyse what you're doing when it feels bad. Can you do the funky chicken mid-corner? Could you go round some of these corners with just one hand (Your throttle hand!) on your bars and the other in your lap? (assuming you don't need to feather your clutch). Where
exactly are you looking when it feels bad? At the road ahead or down on to the road surface staring at all that slippery stuff?
You
should have the lightest touch on your bars, but the reality of stress (wet winters morning, with cold tyres, a slippery urban road and an HGV in lane two) is that it manifests itself in us, with i/ tight arms, ii/ chopping the throttle mid corner, iii/ very low vision / target fixation (staring at every manhole cover and squashed fag packet), a tense torso, and the worse sin of all, grabbing the brakes mid-corner out of sheer panic.
Do you recognise any of those symptoms ??

My body still wants to do that when I'm stressed! Yes, my stress levels may not kick in as earlier as yours - but it's all relative - they'll still kick in if I push the envelope hard enough. But the power and the knowledge I have, means that I
understand why my brain does what it does. And that helps me cure it.
So if I'm steaming into that shiny bitumen, badly cambered bend, maybe I'm in a bit hot, I know what will fire up my 'bad triggers' if you like, and what my brain will try and do to me. So in return, so I'll counter them back.
Tight arms will see your steering just go to pot. Imagine a steering damper on your bike. Well you have two!! Yep - both yer arms! And when you get stressed, and they tighten up - yer bike won't steer - it'll go straight on. Target the tension in your arms by making yourself relax. Check your elbows, do the funky chicken, sing 'I feel like chicken tonight ... '. Don't suffer with tight arms. (A good excercise on a country road is to stick the bike in 5th and drive for mile after mile with your left hand in your lap..

)
Target fixation is real. It happens to us all. If we're on a wet road, and we're about to ride over a squashed fag packet or coke can, will it actually throw us off our bike?? No, it won't. I promise you, you'll just go over it with no drama at all. So why look at it ?? It saps out attention, it kicks in tension

eek Gasp ..... fuck ..... man-hole cover coming up ..... tense arms and stare at it ...

..) and staring at it, will cause you more problems than just riding over it. Recognise it in your riding! Tell yourself not to stare at things (especially the floor) looking at what you're riding over will not help you one jot !
Finally, counter steering ...
Work out what works for you. Some peepes like to push, some like to pull. So don't be a slave to what you read. Have a play with weight through foot pegs. I do a fair bit of off roading, and for me, the very scenario you're talking about would see me weighting my outside peg, and
lifting (pulling) that same outside bar with high elbows ... against the weight of that outside peg. (Geddit ??) Almost like putting a foot on a door frame and pulling the door handle of the jammed stuck door. Some may say that's a load of shite

, some may say they prefer pushing a bar ... it doesn't matter. What matters is what makes you feel confident, and what works for you.
Go and play. Play games. Ride standing up, ride with one hand, ride with no hands, ride a country lane standing up .... then with just your right hand ....
Learn the relationship of how you, as a rider, when you're stressed, restrict and ultimately fuck up the bikes ability to go round a corner
