Your defrost heater should not read 0 ohms.

We had the same problem with ours (2002 year, same model). Called out the Sears repair man, and he told us the problem is with the control panel (or board). He said the freezer is running too cold, so that the drain is freezing up (even though it's not clogged). The cost to replace was over $600. You can order the board directly from Sears (http://www.sears parts direct.com/kenmore-refrigerator-parts/2216920/0046/106/model-10652703100/0582/0165000. html). I haven't done this, yet, so I'm not sure this is the cause. I'm simply putting out here what the repair man stated.