A few clarifying questions and then a general piece of advice.
- How much ammonia are you dosing with your fishless cycle (in ppm)?
- How many live plants are in your tank? Light, etc.?
- What is your substrate?
- Did you transfer any media from the older tank to this one?
Finally, Stability is not the same bacteria as you actually want to have in your tank, though it will process the ammonia. It will actually compete with the ones you want, unless there's been a recent formula change. Tetra Safestart and Dr. Tim's One and Only are the only products I am aware of that contain the proper bacteria species that we are trying to encourage the growth of in the tank. These attach to surfaces in the tank and become native. Most of the others are free swimmers... and don't actually stay there long term... that's why their product suggests adding it so frequently, whereas the ones I mentioned are a single addition at the beginning of the process and that's it. Stability does offer a wider range of bacteria than those I mentioned, which may benefit other portions of the natural cycle, specifically anaerobic bacteria, but I don't think it works to be a long term solution. By its own admission, it requires additions daily. An additional reason I recommend against Stability and in favor of the other two brands is that Stability claims to be effective in both marine and freshwater environments. But, the bacteria which are proven to work in freshwater systems, aren't suitable for marine environments, and vice versa. It may be a great product for marine, but for my money, its just not as effective as the other two (here's the disclaimer though... as long as the product has been properly handled - kept within proper temperature parameters during shipping and storing before it was placed on the shelf at the store).
Now, for a couple of general comments:
It would be far better to move the cories over to the 10 gallon tank, rather than the snails. And it would be best to get a few more cories. They are shoaling fish, and they do best in large numbers.
Catfish, in general, are not a 'clean-up' crew. Cories, in particular, have a very specific diet and do not eat pest snails. They require 'bottom dweller' pellets, and these should include meaty, rather than algae, ingredients. They are also rather active fish, and as such, a 10 gallon aquarium is an absolute smallest tank that they should be kept in. The larger species of corydoras require more room. The dwarf species of corydoras can do nicely in a 10 gallon, in proper numbers. They also prefer a soft substrate, something like sand or similar. They sift through the substrate with their noses and barbels and actually filter the sand through their gills looking for their food. (Its very cool to watch them do it!) Gravel, on the other hand, doesn't really offer them that same opportunity and makes scavenging harder for them, and can lead to them getting small scrapes on their 'snout' and barbels, which can lead to infections.
If the snails are the main object of your affection, try to rehome the cories, with a friend who has a tank, or some LFS will even take them back. And for the 10 gallon tank, you could put both the snails and some colorful mid-water swimmers like Trigonstigma hengeli or espei. (commonly called rasboras - hengeli or lambchop.)
And finally, welcome to the forum!