The phrase "really carefully" is correct and usable in written English.
You can use it to emphasize the degree to which something is done. For example, "She prepared her proposal really carefully."
Others listen really carefully.
I cast really carefully, and then I join the club.
They're not just mucking about, it's really carefully done.
"I want you to listen really carefully now," he said.
To Mr. Coddington this indicated that these canvases "were really carefully conceived compositions".
Over the years it made me choose friendships wisely and really carefully.
"You have to look left, look right, really carefully," Mrs. Bifulco said.
"We had to hang the old ones really carefully," she says.
We planned really carefully, especially after losing to Celta de Vigo [in their penultimate league game].
Sometimes learning can be tricky, and we get stuck if we don't listen really carefully.
We really carefully balance the exciting, the innovative and the new alongside the popular and successful.
