Do languages in $\mathsf{coRE} \setminus \mathsf{R}$ have Turing machines?
We can associate a language to a Turing machine in several ways.
If the Turing machine halts on all inputs, then the language accepted by the Turing machine consists of all words which cause the Turing machine to halt in an accepting state. The class $\mathsf{R}$ consists of all languages which are accepted by some Turing machine.
For an arbitrary Turing machine, the language recognized by the Turing machine consists of all words that cause the Turing machine to halt (in any state). The class $\mathsf{RE}$ consists of all languages which are recognized by some Turing machine.
If $L \in \mathsf{coRE} \setminus \mathsf{R}$, then in particular $L \notin \mathsf{R}$, and so no Turing machine accepts $L$. If $L$ were recognized by some Turing machine then $L \in \mathsf{RE}$. However, this is impossible, since then $L \in \mathsf{RE} \cap \mathsf{coRE} = \mathsf{R}$.