Russia has always had plenty of cockroaches - just different species live in different places. American cockroaches (Madagascar hissing cockroaches and other large species) are more of a rarity in our climate, and there's no mass invasion of dachas happening yet. Sure, sometimes they show up in Russia through imports or greenhouses, but that's more of a rare occurrence than a trend.
People exaggerate the danger. They don't bite, they're not venomous, and they carry diseases no worse than regular cockroaches (meaning they can be carriers, but that's not specific to them). The main thing is they're just unpleasant and can eat plants and damage food. Outside, on a dacha, American cockroaches honestly don't do well in our climate and don't breed actively. If they do show up, it's most likely hitchhikers from a greenhouse or some escaped pet order.
If they're already on your dacha, regular methods work: clean up food scraps, don't leave damp spots, use traps or insect gels (the same ones you'd use for German cockroaches). The main thing is don't panic and don't create conditions where they could survive winter. The cold will get rid of them quickly, so in our climate this isn't a constant threat.