The tires you have are OK. Pump them to a low pressure and take it easy. Fatter tyres might be better on ice, but worse on snow (won't plow their way to the pavement through snow). Then again, in deep snow, fatter tyres with big grooves are better.

Getting a pair of studded winter tyres will cost about 70 euros, lasting 5 or more years since you don't have room. I prefer that expense, to falling and breaking bones, or not riding, or riding at walking speed.

If you have room, get a spare set of wheels with winter tyres.

Even better - get a cheap (i got mine for zero euros) mountain bike, fit it with spiked winter tyres and ride it when it is snowing. Salt can be bad on bike and components, so it pays in the long run to have a winter "beater" bike.