Your soil should feel dry at the surface within a day or two. If it's taking weeks, then you need to repot with a more porous soil. You probably have root rot which is why it's not growing. The wet soil could also be rotting the lower leaves that are resting on it. I would remove all of the old soil from the plant, trim away any rotten roots, and repot using a gritty soil. I did this to an Aloe Vera that had the same problem a couple months ago and it hasn't grown yet, but hasn't looked any worse either. Recovery may be very slow during the winter with few roots, but with time it should come back better than ever. Try to give it lots of light since that can be a problem indoors. Here's a link to a very good soil recipe. I've found the specifics are not that important as long as the particles are all large enough that water runs right through them. If you don't have the time or patience to find the ingredients, at least find one gritty component and mix 1-2 parts of it with each part new potting soil. Here is a link that might be useful: Succulent soil mix

Live dangerously. Re-pot it, deeper, and cover the bare stem with soil. Perhaps it will grow roots from the stem and be better off. Some plants can take planting more deeply and others can't. And some you think couldn't possibly cope, but then someone tries it and... VOILA! I know of a case where some pot-bound Eucalyptus were planted very, very deeply (up to their necks) along a river bank. They were tubestock, left in their pots for too long, and should have been thrown out. However, it is very good that they were planted by someone who wanted to give it a go, because now they are stabilising that river bank. The covered stems grew roots (adventitious roots) and it worked very well. Then they all got married and lived happily ever after.

Well, a bigger pot holds more soil and the soil holds the water. That can lead to rot, and you don't want that. By mixing perlite with your cactus soil, you can improve the odds of the plant not sitting in water. Mix 1 part soil to 1 part perlite and you'll have it cut 50%. I have my Aloe vera in an 8 inch orchid pot. I'm sure it was overpotted at first. But I have used a good draining mix for the plant and it stays outside all year. The top dressing of rocks helped hold it up until the roots took hold well. You can see the outer leaves are starting to yellow and die back, but the middle growth is still good.

I would take it out of that decorative pot and use a saucer instead. I water my aloe when the soil looks dry. If I were you I'd not water it for at least 2 weeks. It is a succulent, so it will survive. With most of my plants I can tell if they need water by lifting there pots up. If your doesn't have drainage you should repot it.