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Did you test the old switch to see if that is the problem? Many times this is a result of a bad connection at the bulb or connections TO the bulb and not the switch. If you have a small digital multimeter we can do some testing first and be sure that this isn't something else. IF NOT, I can direct you in the switch replacement. All follow ups are included in our fee.
I am attaching a link to the exploded parts view on this for you. Part number 41. Is that the part you have? http://www.laurelhurstdistributors.com/parts/refrigerators/dometic/NDM1062_5.html