Can ionic compounds dissolved in organic solvent conduct electricity?
Sure they can.  Even if most of the ions are in ion pairs or other complexes, they can still move by being transferred from one complex to another.  All you need is to get the ionic compound to dissolve, which in most cases requires a polar solvent with which the ions will not react.
A rather unexpected case involves magnesocene, whose bonding is a mixture of ionic and covalent.  When dissolved in ether or THF, magnesocene forms enough ions not only to impart significant conductivity in solution, but also to be proposed for use in batteries:

Unlike ferrocene, magnesocene displays slight dissociation and subsequent ion association in polar, electron-donating solvents (such as ether and THF).[1]
$\displaystyle {\ce {MgCp2 &lt;=&gt; MgCp+ + Cp-}}$
$\displaystyle {\ce {MgCp2 + MgCp+ &lt;=&gt; Mg2Cp3+}}$
$\displaystyle {\ce {MgCp2 + Cp- &lt;=&gt; MgCp3-}}$

Note the single charge on all ionic species.  This is to be expected in organic solvents with their weaker polarity and ion solvation power relative to water.
Cited reference:
1.
Schwarz, Rainer; Pejic, Marijana; Fischer, Philipp; Marinaro, Mario; Jörissen, Ludwig; Wachtler, Mario, &quot;Magnesocene-Based Electrolytes: A New Class of Electrolytes for Magnesium Batteries,&quot; Angewandte Chemie International Edition 2016, 55(48), 14958–14962.  https://onlinelibrary.wiley.com/doi/abs/10.1002/anie.201606448