How are the snails doing? I've found them to be fairly robust to bad conditions, but that is very high ammonia. There are reasons to be skeptical of API ammonia tests, but only on the fringes (like differentiating between 0 and 0.25 ppm). At minimum, take some water to a local fish shop, even Petco/Petsmart usually do tests for free. The fact that the pH is going haywire is consistent in my opinion with the ammonia getting out of hand. Do more frequent water changes until the tests look better.
I'm going to refer to these as mystery snails since I assume that's what you have. Many people use "apple snail" to refer to a closely related species that gets very large (more like a baseball at adulthood), eats live plants, and is highly invasive and is regulated in the US. Mystery snails produce a ton of waste. I've spent time raising these snails and in the beginning it was very hard to give enough water or filtration to keep the ammonia at 0. You can manage 3 of them in a 10 gallon, but I would consider that a heavy bioload and one that will need careful management until it stabilizes.
I suspect the sponge filter is insufficient. You're going to need more water turnover to effectively process the amount of ammonia being produced. You can get a hang-on-back filter pretty cheap that will work in a 10 gallon. I bought a Deep Blue branded 100 gph filter at a local fish shop for about 10 bucks and put my own biomedia in it (I like the porous ceramic types that you can get pre-bagged for Fluval filters) and took out the included carbon.
I would still leave the current sponge filter since it's the only mature thing around. With that said, I believe sustained exposure to very high ammonia levels can have a negative effect on the nitrifying bacteria, so there's a chance if your tests are accurate that the sponge filter has lost much of it's bacterial colony.