Looks like a nice clean original... outside of the refridgeration problems??
did you do the light test on the seals?.. Put a battery operated light (bright lamp works best) inside, close the lids turn off lights and see if you can see any light coming out? if so there is your leak...

   The reason you are getting so much ice build up on the exterior cooling line is because the compressor runs too long.  When it shuts down, the ice will thaw in a minute or so resulting in a small amount of water (put a towel under it ).  I bet that Glen hit the nail on the head. The tube where the thermostat goes into has broken away from the liner and although the thermostat may be working, it can't get an actual reading.  Very common problem with Glasco and Ideal coolers.  I had to drill a hole in the floor of my Glasco (for wiring)- put the thermostat mounting bracket in the pre-cool area, and just have the thermostat  in the cooling cabinet itself.  Quick and inexpensive .  Not 100 % authentic, but it beats pulling the machine apart for my own use. 

    For the record, there are no solder joints. All cooling lines were stuck to the inner liner with tar.