This is what I started with. We need new seals.
This is the bad corner, where the seal leaks the most:
I know, I know. I promise it won't get this bad again!
First, I unplugged the fridge.
I started by filling my "custom ice maker" with hot tap water. I really don't understand why it's called an "ice maker", but it came with it and it's pretty cool looking. I put a few towels on the top of the fridge door to soak up any water that ran out.
I put it in the freezer to use to radiate heat into the freezer area. I've heard people suggest boiling water but that seems a little extreme to me.
Next I used a wooden spoon to lightly scrape the loose ice into the "ice maker". DO NOT, I repeat DO NOT use ANY sharp or metal objects for this. DO NOT scrape hard. You don't want to a damage the inside of your freezer.
I also used a sponge to wipe up water, and dipped it in the hot water to wipe off thin ice on the sides and bottom.
I gathered the loose ice and just dumped it into the sink.
After about 15 minutes, this is what was left
Next, I refilled the "ice maker" with more hot tap water and put it in the freezer. This time I simply closed the door and waited. About 10 minutes later, I opened the door and removed loose ice, then repeated. After 10 more minutes, I did the same. That took care of the ice. I used a towel to "mop up" the remaining water, and I was done.
During this period of time, all our frozen food was chilling out in our equally vintage cooler. It was still frozen solid when I was done.
Reloaded the freezer, plugged it back in, tossed the towels in the wash and cleaned the little bit of water out of the cooler. Done. Easy-Peasy Lemon Squeezy.