Yes, new cartridges should make the problem go away. If it doesn't, you should call Aquatic Life and ask why the product isn't working as advertised.
Yes, since you have the 4 stage it will filter out chloramines. That is probably why the problem didn't show itself in your nano tank until recently.
Just make sure your test kit checks for free chlorine. That is what you are looking for. A pool test kit might work but I would worry they aren't accurate enough. I would stick with the one I linked from BRS earlier. Once you replace your cartridges make a point of routine testing of your RODI starting around the 4 month point. Time to replace when you start seeing a positive test.