Yeast gives off alcohol as part of the process. Different yeasts have different tolerances for how much alcohol they can stand. Basically once the alcohol gets to a certain concentration, the yeast will go dormant. Similar mechanics with beer.
The fact that your cultures doubled is good news. But remember, once the yeast exhausts the available sugars in the flour, they will stop bubbling. That's normal.
If you are leaving your cultures out at room temp, you'll exhaust those sugars quickly.
When I refresh my culture, I leave it out on the counter until it's bubbling and has almost overflowed the container. At that point it goes into the fridge. I will leave it there up to a month before doing another refresh.
Hard to explain but your culture should have a sourdough smell. I can't think of anything to compare it to. Using supermarket sourdough for this may be an exercise in futility as those breads can be pretty lame. My guess is there's nothing wrong with yours.
The only thing I can think that's different from what I do is leaving it out at room temp for more than half a day.
You can abuse sourdough starter up to a point but once you say you're sorry it's your friend again.
Time to make some bread?!