I have a 1990 Maytag rsw24a side-by-side refrigerator. There is
ice on the bottome of the freezer. This started to happen about
three months ago. When the ice thickness reached about 4", I
emptied the refrigerator and defrosted it. The problem went away
until about 10 days ago, when there again was 4" of ice on the
bottom of the freezer. This time, I defrosted the refrigerator and
used a wire to clear the drip tube. Last Friday, before I went out
of town, I checked the freezer and it was free of ice. Today,
however, there was 1/8 inch of ice on the bottom of the freezer. I
emptied the freezer and poured some water into the drain on the
back of the freezer (below the defrost coil). It drained fine,
albeit slow. I estimate it took about half a minute to drain 50 cc
of water. There is a thick piece of black rubber sitting between
the drain and the freezer. There was some water leaking from it.
Not sure how important this is. When there was 1/8 of ice on the
bottom, there was little ice around the black rubber.

The refrigerator side seems to become colder (vegetables become
frozen) when the freezer has ice accumulation.

There is no ice build-up on the defroster. When I removed the back
panel, the defroster was clean.

Although the refrigerator came with an ice maker and ice/water
dispenser, I don't use them. The refrigerator is NOT connected to
a water source.

Also, about five months ago, the two doors became quite loose. It
seems the magnet stopped working. But they cured themself and are
working fine now. Not sure if this is related to the ice problem.

Any idea what's wrong?

Check the gasket. I had the same problem and the gaskets were coming
out of the slots that hold them in.