It's very hard to tell if the 'dragging across the road' is you skidding on ice or snow and so the car not going when you expected it to, or just that if you run along the side of an unploughed bit of snow, it tries to 'drag you in'.
If it's actual skidding, then you aren't driving according to the conditions - I'm a great fan of winter tyres but even they won't let you drive as though you are on a dry or wet road, and not an icy or snowy one.
Is there a tyre that allows you to drive 'normally' in all conditions? No - what with that contradicting the laws of physics an' all.
The standard (ie, summer) 175 width tyres are certainly better in poor conditions than the 205 width ones, but not by enough that the conditions can be ignored.
If you're talking about being dragged off the side of the unploughed/undriven bit of road, then winter tyres of almost any size are amazingly effective - it can be hard sometimes to feel if you are driving on the bare tyre tracks of the cars in front or on the snow either side. 175 summer tyres won't help much compared to 205 summer tyres for this behaviour - it's the tyre type and tread pattern that matters for this.
I suspect that what you need most is not to try to buy a cure, but to learn to drive according to the conditions. Most drivers don't - for example, they just drive at the same speed on wet or dry roads, so they have a huge margin of safety on dry roads and a small margin on wet roads. These folk are best advised to stay home in snowy and icy conditions as expecting to drive 'normally' will lead to little bumps and scrapes where they can't stop or steer like they usually do.
To learn to drive on winter roads, there is just no substitute for practice and being able to make mistakes where they won't result in a crash. Sadly supermarkets and shopping centres nowadays clear their car parks of snow, so they aren't the great training ground they used to be, but it's worth hunting out other big car parks to practice on - I can think of quite a few on the edge of the city here where folk go to walk their dogs and those are generally empty in snow.
The trick is to learn to feel when you have reached the limit of grip (you seem to have got there already!) and then practice regaining grip. I would say the single most important thing is to learn that stamping on the brake pedal is the wrong thing to do 99% of the time as that guarantees a skid and once skidding you aren't deciding where the car will go.
I had a Norwegian spend maybe ten minutes 'teaching' (mostly complaining at my ineptness) about snow driving and the bit I now find instinctive is that when I start skidding, I disengage the clutch and don't brake - this means that at least neither the engine nor the brakes are trying to make the tyres slip and so gives them their best chance of steering where I want to go.
And remember, cars don't skid on their own - it's the driver wot done it.