I have an Amana Model # ABB2223DEQ with a bottom freezer.
Same problem, water over flowing the ice maker into the ice cube bin below. Water was also getting frozen in the tube that runs to the tray from outside rear of the refrigerator. I have well water and my saddle valve was tapped into the water supply upstream (before) the water-softener. This was allowing hard water minerals to get into the valve behind the refrigerator and the ice maker itself.
So I replaced the line and did drill a new hole 1/8" drill bit into the water supply downstream (after) the water-softener. Hand drilling is necessary (shut-off house water first) because on the back of the refrigerator it says explicitly not to use a self-piercing valve (to make the hole). The hole can be better made manually with a drill and it will be less prone to clog, but it's actually ok to use the valve with self-piercing tip but after you have drilled the hole using 1/8" drill bit.
Then I tried the ice maker but still had the same overflow problem. I removed the ice maker and used a long screwdriver to make sure that the feed tube had no ice blockage. Then I removed the ice maker and melted ice blockage in the ice maker where the tube entered it using warm water and cleaned the whole thing. My last ice maker had a ton of hard water (white powder/scaling/paint chipping) deposits all over it. This fairly new one still looks pretty good. Reinstalled it. From outside of the freezer each time, I watched to make sure feed tube went into its proper slot.
I tried ice maker and still had water overflow problem. Finally, yesterday I stuck my head all the way into the back of the freezer and checked refill tube placement... it was wrong. I realigned it and voila, it works perfectly now.
That tube was the big culprit. (In side-by-side freezer-refrigerators I suggest getting a mirror and carefully examining the feeder tubes' placement.