When an aloe plant is being overwatered, the leaves develop what are called water-soaked spots. They look like what you describe: soggy and soft. It is almost as though the entire leaf becomes saturated and gel-like, then it turns to mush. Eventually the entire plant dies. I think that your plant is experiencing a waterlogged condition because the pot you put it in lacks a drainage hole. Avoid planting in a pot without a drainage hole. The layer of pebbles in the bottom of a pot compounds the problem. As moisture moves down through the soil, it forms what is called a perched water table over the pebbles. Not until the soil above is saturated will the water move down into the pebbles. That means your aloe's roots are constantly saturated. The soil is waterlogged, and the plant's roots are dying from lack of oxygen.
You might be able to save your plant if you dig it up and let it dry out for a day or two. Remove any leaves or tissue that appear to be dead. Then dust the dry base of the plant with rooting powder and replant it in a pot with a drainage hole. Give aloe bright light, and keep it on the dry side. If you want to use a beautiful pot that has no drainage hole, drill a hole for drainage, or use it as a cachepot. Tuck your plant into a plain plastic pot that can fit inside the eye-catching container. Elevate the inner pot on 1/2 inch of pea gravel.