It turns out we have/had multiple problems. First, the fins were iced over and dripping down outside the drip pan. We had filled the fridge too full and had he door open too long I guess. We were able to clear a lot of that and it helped.
Second, the ice maker line had two pinhole leaks. The cheapo plastic was pretty brittle so we put some of that rubber-sealing tape stuff and that is now a slow seep. We will need to get that replaced but I bet it will be expensive from what I've read. I can't physically pull the fridge and will have to pay to have that done.
Third, the drain lines from the drip pan were so crunchy brittle that would fall apart when you touched them. I'll need to get those replaced and resealed.
Thank you all for the information. Between your advice and the internet I now know at least what is wrong. I appreciate you.