Are there unary languages that are in RE but not R?
There are undecidable unary languages.

You can consider the set of words in $\{0,1\}^*$ as essentially encoding natural numbers in binary.*  Similarly, the set of words in $\{1\}^*$ encodes natural numbers in unary. Intuitively, if a set of natural numbers is undecidable, that doesn't depend on the base in which the numbers are represented, since we can easily write algorithms to convert between bases.

Another way to see this is to observe that any decidable language (over any alphabet) is, by definition, decided by some Turing machine.  There are only countably many Turing machines, but there are uncountably many unary languages.  Therefore some (almost all!) unary languages must be undecidable.



* This isn't quite true because of the empty string and because, e.g., the strings $1$ and&nbsp;$01$ are clearly different but both encode the number one in binary. To get around this, say that the string&nbsp;$w$ encodes the number whose binary representation is $1w$.