Need Appliance Repair Help? Ask a Technician for Answers ASAP
The light pulsating has nothing to do with the other issue of the ice melting and re-freezing. The light issue is either a bad connection with the bulb in the light socket, or a bad door switch contact. Try another light bulb and if you still have problems, you'd want to replace the door switch.
As for the ice melting and re-freezing, that is typically caused from the refrigerator staying in the defrost cycle for too long. There is a 1" long white plastic bullet sensor called a "thermistor" clipped onto the top of your silver evaporator coils which is supposed to tell the refrigerator when to end the defrost cycle if it's getting too warm that sounds like it's failed. This is mounted on the top of the silver evaporator coils which is located behind the lower back panel INSIDE the freezer. You'd need to remove the panel to expose the coils and see the thermistor to replace it. Below is a link to the part you'll need to order for your model.
CLICK HERE for thermistor
We are on the honor system here. Please remember to rate my service by selecting a star at the top of the screen before you leave today. If you need anything else, just let me know. You can continue asking follow-up questions at no additional charge after rating my service if you still need assistance.
Hello again. I'm just following up because I haven't heard back from you and I see this question is still open. Is there anything else I can help you with?