Going brown/purple is what Aloes do when stressed - this can be as simple as temperatures going down in winter and usually nothing to be worried about. It happens, and in most cases they will go back to greenish when the conditions get better.
The going squishy is the problem here... that means the plant is rotting. I realize you needed to repot because it was in bad soggy soil, but normally I would avoid that when it is cold. The plant is much more likely to not take it well, especially if roots get damaged - a perfect time for something to start affecting the plant in a negative way. But if it has to be done it has to be done, best practice then is to make sure to leave the plant bareroot to dry out for a few days - any damaged areas can dry out and callous over - before repotting, and then not watering for a week or more to give the roots some time to recover.
So I think the watering right after repotting (if I understand your post correctly) is probably what is giving it, its current problems.
Given that it is a big plant, I would unpot again, inspect the roots, if anything is obviously rotting cut that away. Cut all the squishy parts away, or alternatively if squishy is more prevalent than not cut off all the healthy sections and let those dry out in a warm and dry place. Then get a fresh pot of well draining soil and replant. Then leave it alone no watering for at least two weeks, it will take a while to recover. Most Aloes do not like wet feet at all when it is cold, so I would be very very sparse with watering Aloes as a general rule, until it gets nice and warm outside during the day and it does not get very cold at night anymore
