No, woolly mammoths were not alive during the same time as dinosaurs. Woolly mammoths existed much later, during the Pleistocene epoch, which began around 2.6 million years ago. Dinosaurs, on the other hand, roamed the Earth during the Mesozoic era, which lasted from about 252 to 66 million years ago. The last of the non-avian dinosaurs went extinct approximately 66 million years ago, long before woolly mammoths evolved.
This means there is a significant gap of millions of years between the extinction of dinosaurs and the appearance of woolly mammoths. While they both are fascinating creatures in their own right, they lived in entirely different geological periods and environments.