A limited slip or
locker will also spin both tires. The problem with an open differential is that all the power is sent to the wheel with the least amount of traction, so you'll find yourself stuck spinning one tire while the other just sits there.
I would recommend a limited slip differential for your application. They're more pricey than locking differentials and can be more costly to install, but it's transparent onroad as opposed to the quirks of a
locker.