Nibbles, 7 bytes (14 nibbles)
/|``p$/!>>$@!*
Systematically (non-randomly) finds all permutations of the input without adjacent identical nuts, and then returns the first of these (or an empty list if there aren't any).
/|``p$/!>>$@!*
| # filter
``p # all permutations of
$ # the input list
# by
/ * # fold-by-multiplication
! # zip together
>>$ # list without first element
@ # the same list
! # are elements equal?
/ # finally get first element
