If you have a continuous sweating problem I would also look at the seals
around your doors. This will let in moist air and everything will always be
wet.
The defrost elements should be available if you have a "worth fixing" type
refrigerator. My old Kenmore was 17 years old when I bought my house and I
have put on new door gaskets and a defrost timer. Which you should check to
make sure isn't staying on. The defrost timer is probably the #1 item which
goes bad and people simply go buy another refrigerator instead of fixing it.
My defrost timer was under 20 bucks and the local appliance repair center had
one on the shelf.
If I were to place a bet on what happened in your fridge it would be a bad
defrost timer. Your fridge iced up which broke the wire. One simple check you
can make to determine whether an appliance is even worthy of being fixed is to
take an ohm meter and check the cord prongs to the case of the appliance. If
the hot or neutral wire read anything to the case of the appliance it should
really be bye bye time for that item. Compressor motors are bad about high
resistance grounds, they won't trip a breaker but if you get between the case
of the appliance and ground it is more than enough to kill a person.
One other incentive to replace is the efficiency of refridgerators even made
10 or 15 years ago is low enough that a new one will begin to pay for itself
in saved electricity.
In your case any fix that might not properly insulate this wire is like
loading a human mousetrap. If you can't properly fix the defrost element or
can't find a new one its time for a new fridge.

tim