Is "uncarefully" a word?
Solution 1:
No, it isn't. Carelessly is the word you are looking for.
Solution 2:
Carelessly is much more common, and, generally, you should use it, but, yes, uncarefully is a word. Google Books shows it was used by Richard Bulstrode (died 1711) and has been in use ever since.
Solution 3:
For an alternative that, to me, has a slightly different connotation from "carelessly", how about "incautiously"?