Sounds like the drain tube is plugged.

It runs from under the evaporator coils in the freezer to the drip tray under the unit.
Some units drip into a funnel like fitting at the top, back of the fresh food section and then down to the drip tray.

Can be cleared with a turkey baster and warm water, compressed air or a piece of wire.

Check that the drain tube is not clogged with dust etc. at the drip tray.
You might as well clean the condenser/compressor fan and the coils while you are down there.

If you see stuff growing in the tube flush it with a 10% bleach solution, let it sit for 15 minutes and then do it again. After another 15 minutes flush with clean water.

Be careful that the drip tray does not overflow and emty the drip tral after you are done.

Be sure to unplug the unit while doing this.