This should not be happening if the hub bearings are adjusted properly. The cones are threaded onto the axle so their spacing can be adjusted by turning them on the axle threads with a thin wrench (called a cone wrench). When spaced properly they let the wheel spin freely with no noticeable play no matter how tight the QR is adjusted. Sounds like yours are adjusted too tight. As already mentioned, the traditional type of QR (internal cam with sharp knurled teeth to bite into the dropouts) should have no trouble keeping the wheel from slipping. Another potential problem is if the end of the axle protrudes too far past the nut on the end. In that case the QR might be clamping down on the end of the axle rather than exerting all its force clamping the dropout.

There should be no need for the additional complexity of a separate chain tensioner.