Goldfish are tougher because they have scales and gill covers, where axolotls do not. Your goldfish tanks have certainly cycled, it's just happened without your knowing it
The guy at the store was right though, they do need the same basic conditions as goldfish- COLD, clean water. You need to get the temperature under control somehow. Axolotls can't handle heat for very long without succumbing to the stress. After only a week and a half, I don't think they could be starving to death. More likely they're too stressed from heat and poor water quality to feel like eating.
I don't think you're understanding just what cycling is. You DO NOT need to do a full water change in your tank. Get the axolotls out of there and leave the tank alone. The only water you'll need to change daily is the water in the axolotls' temporary housing. You DO NOT need any products to lower ammonia and nitrite levels. Nature has given us her own solution- nitrifying bacteria. But if you don't stop fiddling and let the cycle complete, you won't ever build up a big enough bacterial colony to do the job properly.
Just a quick review of what happens when a tank cycles:
1. Fish/axolotls produce waste. As this waste decomposes, ammonia is released into the water. Ammonia is toxic to living things.
2. Bacteria in the water called nitrifying bacteria feed on the ammonia, breaking it down and releasing a new toxin, nitrite. As they feed, the bacteria are also multiplying. Now there is no ammonia, but we still have poison in the water, nitrite.
3. The bacteria continue feeding, breaking the nitrite down into nitrAte. Nitrate is pretty safe, only very high levels are potentially harmful.
4. Your fish/axolotls have been producing waste this entire time. The process of converting ammonia to nitrite, then to nitrate continues. At some point, usually about a month into this, the size of the bacterial colony has grown so that it is consuming ALL of the waste. At this point, when you test your water, you will find ZERO ammonia and ZERO nitrite, because there is no excess left- the bacteria are getting it all. This is when your cycle is complete.
5. NitrAte readings at this point will usually be around 20-25ppm. This is ok, safe. Since there is nothing feeding on the nitrAte, these levels will rise as more and more ammonia/nitrite are metabolised. Water changes are meant to control the nitrate levels in a tank. By taking high nitrate water out and replacing this with freshly treated tap water, we reduce the overall nitrate levels in our tank. We only do water changes once the cycle is COMPLETE. Changing water before this slows the process, as you're also removing the ammonia/nitrite that nourish the nitrifying bacteria. Once the cycle is complete, weekly changes of 10-25% are enough to keep a properly stocked tank healthy.
***6. Instead of leaving your fish/axolotls in a tank while it cycles, it's more humane to replace the waste they produce with another form of waste- food. Fish flakes, pellets, a piece of shrimp... basically any organic kind of waste. You could also add ammonia directly in the tank. I've never tried this myself, but have heard you can buy it bottled and ready to use.
We (aquarists) call this "cycling" but if the process seems familiar it's because you probably learned it in high school biology, under another name: the nitrogen cycle.
Hope that helps a bit
A better filter is a good idea though- sponge filters are usually junk. An external or internal filter with a large chamber to hold filer media (sponge, sometimes carbon) will give the nitrifying bacteria lots of room to grow (they end up living in the filter media, creating a "biological filter" within the mechanical one.) You could also look at an undergravel filter, a lot of people here use them and have found that their axolotls do really well with them. When you get your new filter, look for something that's not too powerful and has an adjustable flow. That way if it's a bit too turbulent you'll be able to slow the flow a bit.
