I'm not sure what this succulent is called, but pretty much since I got the plant which was many months ago the leaves are always soft. Even several days after watering.

I thought maybe the soil is hydrophobic, so when I water it I sit the root ball in a bowl of water for several hours.

I've put a moisture probe into the root ball, almost to the center, and it reads as being wet, so either all of the root ball or the majority of the root ball is wet, yet the leaves still remain soft.

Does anyone know what is going on here? And will it be okay? Thanks.

The pot is too small for the plant, the soil looks like it could use better drainage, and it looks like the plant would enjoy stronger light. Your core observation that the plant does not "pump up" after watering would seem to suggest the roots are not functioning well, or the stem is rotten near the base. I wouldn't rely on a moisture meter to tell me anything unless I could confirm its reading by some other independent observation. Maybe avoid sticking that in the root ball if you suspect the roots are somehow sick or damaged. Have you wiggled the base of the plant to make sure it's still attached to the soil?

That Echeveria is a bit prone to melt down, so maybe now would be a good time to (very carefully) remove the biggest offset and use it to start another plant for backup.

I would recommend a pot about twice as wide, with soil that's about 50% rock (pumice/perlite/gritty equivalent). I realize you can't always provide strong light but maybe with the warmer days you could indulge your plant with some outdoor sun (after a careful, gradual introduction, of course). A larger pot will help you out in a few ways. One important way is it will provide a surface for water to evaporate out of the substrate. Another is it will provide you an easier place to water (watering the soil, not the plant).

If you really want to water to completion (which I generally do) the easiest way is to water in 2 or more passes, separated by a few minutes each time.