OK, chances are that the fish are suffering from ammonia poisoning.
All fish produce ammonia, mainly from their gills. Ammonia is highly toxic. In order to combat it, your filter grows a colony of bacteria which eat ammonia, and turn it into nitrite. Unfortunately, nitrite is also highly toxic. To combat this, your filter also grows a second colony of a different bacteria which eat nitrite, and turn it into nitrate, which isn't that toxic in sensible concentrations.
Whoever told you that you didn't need a filter for your betta was talking out of his rectum. I'm quite surprised that your betta hasn't died. However, with one fish in a relatively large tank, the ammonia wouldn't have built up that quickly, and a weekly water change may have been enough to control it. Also, there may be some bacteria in your gravel that has helped.
What has happened now is that instead of one fish, you have 3 fish, so the ammonia builds up three times as quickly in the tank.
Another problem is that the outflow from the filter is the main way oxygen gets into the water. Without a filter, the water may well be lacking in oxygen. Again, bettas are naturally used to living in an envrionment with a lack of oxygen, and are naturally adapted to deal with this, but GAEs certainly aren't.
I strongly recommend you invest in a sensible sized external canister filter, and thoroughly read the green link in my signature area. It will tell you how to get those bacteria colonies in the filter growing. In the meantime, you need to be doing massive water changes, every day, to keep ammonia under control.
The 2nd thing you need to invest in is a set of liquid-drop water tests, for the 4 parameters that I mentioned above. You will need these to work out how much water you need to change whilst your filter grows the bacteria. I personally recommend the Salifert or Nutrafin tests, although most people on here recommend the API. Don't get the paper-strip tests, they are not accurate enough for someone in your position.
The good news is that your acclimatisation procedure is spot-on perfect!
Any further questions, fire away!
