Ore extensions of commutative rings and the Dixmier-Moeglin equivalence

We consider Ore extensions of the form $T:=R[x;\sigma ,\delta ]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)$ is primitive if and only if $\{P\}$ is locally closed in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)$, if and only if the Goldie ring of quotients of $T\mathrm{/}P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.