The shop is talking rubbish I'm afraid.
Ammonia in water exists in 2 forms, ammonia and ammonium. The only things that decide how much is in each form are the temperature and pH. Duration of tank has nothing to do with it.
In one way, though, the shop is right about not doing water changes because the bacteria will grow faster the more there is. But it will also kill or damage your fish. Cycling with fish is a compromise between keeping the fish safe and having enough ammonia to grow the bacteria quickly. Being exposed to ammonia and/or nitrite shortens the life of a fish, and makes them more susceptible to disease.
The simplest way to deal with ammonia or nitrite is to do frequent enough water changes to stop either of them ever reaching as high as 0.25. That will keep the fish safe, but also make the cycle last longer. As I said, it's a compromise.
But if you don't mind something a bit more complex, there is another way to look at ammonia. Our test kits measure ammonia and ammonium combined; they can't tell the difference. But we can calculate how much of each makes up that reading. All you need are your ammonia reading, the pH of the tank and the temperature of the tank. Then go to
this website.
Enter the ammonia reading in the total ammonia measurement box; set salinity to zero; then your pH and temp in the next 2 boxes. (Make sure you use the right temp scale!) Then click calculate. The figure you want is the lower one, NH3 concentration. This is the amount of your reading that is in the toxic ammonia form. Your aim is to keep that figure below 0.02.
Using this calculator it doesn't matter how high the test results get as long as the calculator figure stays below 0.02 and the test reading stays below 2.0 - ammonium isn't non-toxic, it is less toxic and above a test reading of 2.0 even ammonium starts to become toxic. This will allow your total ammonia-plus-ammonium to get higher than the 'keep below 0.25' way and allow the tank to cycle faster.
But you also need to keep nitrite below 0.25 as there is no less-toxic form of nitrite; it is all toxic. If your nitrite reading gets higher than it is now, there is something you can do; ask if you need to know.
Ammonia treatments can help but only short term. Their effect lasts around 24 hours. You could try using a dechlorinator that detoxifies ammonia, but also do water changes when the calculator says you need to. The tester measures the detoxified ammonia the same as non-detoxified except for when using Seachem Prime. For some reason that one can distort the ammonia test
I'm afraid that the reason your fish are dying is the ammonia and possibly nitrite, if that has ever been over 0.25. Gasping at the surface is the classical symptom as both affect the fish's ability to get oxygen from the water so they go where the oxygen is richest - where the water makes contact with air. If they were getting sick from another cause you would probably see physical signs eg white spots or white patches on the skin, or swollen bodies.
If the plants are growing well, they will help as plants use ammonia as fertiliser.
So, after that essay -
Use the calculator to work out the ammonia portion of the test reading. If that figure is now or becomes over 0.02, do a water change.
Keep an eye on the nitrite reading and if that reaches 0.25, come back and tell us.
And ignore the shop.