It was probably cooler then. And besides it was 12,000 years ago not 10. Around 12,800ya there was probably a major northern hemisphere bolide strike, see the book posted by SC. Then for around 1500 years there was a sharp cooling called the Younger Dryas. Mammoths may have been flash frozen by the bolide strike, even with undigested plants in their stomachs. After the YD ended there have been periodic warmer and cooler phases, but not like the YD.
Your excuse doesn’t make any sense because the flash frozen animals are in the same state today. It’s not the cooling you need to find an excuse for but rather the warming that provided the food for these animals just prior to them being flash frozen. It was warmer then than it has been afterwards or the animals would have decomposed. You’ll likely find that the freshest seeds and grasses preserved in the permafrost is the same age as these preserved animals.