Your drain tube may be stopped up with ice at the upper end because it drains too slow because it's stopped up at the lower end in the evaporator pan under the unit at the floor. It can get dust and mold in it. Once you get the ice out at the top a little pressure with a turkey baster will usually clear it out. Flushing it out with hot water and clorox may help.
Make sure it drains quick enough to prevent refreezing. . The drain should be located below the evaporator coils on the lower back of the freezer.