It does not look good. What appears to be going on is the bases of the leaves are rotting and probably the stem nearby too. Once the rot gets into the stem (at least near the base) there is not much you can do except maybe try to harvest a few leaves and attempt propagating it that way (which tends to be very hit or miss, more likely miss if you've never tried). Don't give up prematurely, this is just my best guess at the moment.
If you put your Echeveria out into outdoor sun, that's going to stress it out if it hasn't already been outside for a while. I would recommend against shuttling these plants outside for more sun, and instead just occupy the brightest possible spot available indoors. Like right in front of your sunniest southerly facing window, assuming you're in the northern hemisphere. It's really hard to give too much sun indoors, assuming mild temps and reasonable air flow. Outdoor sun is another matter because it has a lot more UV in it, and plants need some time (and a gradual stepwise progression over weeks, ideally) to get used to it.
Sometimes these meltdown events with new plants are not related to anything you did, instead relating to some care issues at the point of sale, and only popping up afterwards. It sounds like you've been pretty careful. If you're bottom watering, you should see visually with a clay pot when the pot is saturated with water (it turns a darker color), and that should help you work out the timing. I would typically go for longer (like 30 minutes) but clay pots are thirsty things. I don't see any advantage for bottom watering compared to careful watering from above in this situation. As long as you make sure you don't get water in between the leaves (so water the soil, not the plant) you should be good. Blow out any water that might get in there afterwards. Be aware that you will have to water in more than one pass if you do it from above. So pour some water on, then wait 5-10 minutes for it to percolate into the soil and the pot, then water again. More on that subject here:
The thread "Sunday afternoon experiment: watering in multiple passes" in
Gardening Ideas forum