The Japanese leadership did not see the atomics as significantly worse than what they had already suffered due to the sustained bombings their cities had endured in which many more civilians died than from both the bombs combined.
Militarily, this would be false. Where before it would take swarms of US bombers put at considerable risk to firebomb a city into almost nothing (yet not really damage hardened bunkers/buildings by too much), Hiroshima and Nagasaki showed that it only took one bomber (though both flights had three - including two observers) to wipe out an entire city (or whatever), and obliterate nearly *everything* in it, all with less effort than a reconnaissance mission.
Also, consider that after Hiroshima, most Japanese leaders either considered it to be a lie ("...no, there must have been hundreds of bombers, not three!", didn't know about it fully, or thought it was some sort of fluke bombing that turned into a conventional firestorm, and the survivors were just too dazed to know quite what happened (after all, Hiroshima was completely untouched up to that point). By the time they realized that maybe Hiroshima did happen the way it did, Nagasaki blew up the same way, and survivor stories corroborated way too nicely.
But as for the whole Russians thing? Two reasons why that's not really as consequential as some would like to believe:
1) Japan was doing all of its negotiating through the Soviets, and knew them well enough that they not only didn't care, but were way too busy with cleaning up (and consolidating power in) half of Europe. Even if the Japanese thought they were serious? Siberia worked both ways, and logistics would prevent the Soviet army from doing hardly anything - by the time the Sovs could bring anything substantial to bear, the US would have long since bombed Japan's home population, infrastructure and culture into literal non-existence with these new atomic bombs.
2) From the Japanese militaristic point of view, the US wouldn't even have to invade - but simply stand back, bomb everything (and everyone) into impotent ash, then forget that Japan ever existed. It would be a death without honor, which was a living hell to the Tojo faction's mindset... to be treated as little more than hornets killed with a can of bug spray.