Seemingly unrelated, but very related at the core:
How do you spot oil slicks on the open ocean?
Answer: You look at the waves!
This is what happens when SAR imagery from planes or satellites is used to spot illegal oil dumps. The trick is, that the radar reflectivity of the sea surface depends on the presence of surface waves of a specific wavelength. And the oil slicks look dark in SAR images, because they dampen the short waves that are abundant everywhere else on the ocean. This works even with the thin contaminations that result from ships cleaning their tanks.
So, if a thin film of oil can sufficiently dampen waves with a wavelength of several centimeters, it stands to reason that a thick floating island can easily dampen waves of many meters in length.
Of course, the shore line of your floating island will go up and down with the waves, but if you go some hundred meters away from the shore, all but the largest waves should be entirely dampened out. The total dampening of a floating island should actually be more than the dampening of a rock island, because the floating island won't reflect the waves. It will actually absorb them and turn them into heat.
Tsunamis are no threat to floating islands, by the way: Tsunamis are waves with extremely long wavelengths. They only become dangerous when their wavelength is reduced by the low water depths near shores. As long as there is enough water beneath a floating island, it will just move a bit up and down when a tsunami passes underneath.
Storms are a big problem for your floating islands, though. Hurricanes have immense power, and they will tear at the trees on the floating islands. And with the high equatorial temperatures, you have to expect many strong hurricanes. The hurricanes will threaten to tear your floating islands apart.
You might work around this problem by suggesting that the floating islands consist of plants with extra long horizontal roots, which actively entangle themselves with each other. They actually evolved to stabilize the floating islands. If you have a several meters thick layer of entangled roots, you should get some robustness against even hurricanes. This should work out, since the forces exerted on land by storms are similar for nearby points: The whole island will feel an enormous amount of force acting in a certain direction, but the forces trying to tear the island apart will be orders of magnitude smaller.
The greatest danger to your floating islands, however, is that they will drift between different climate zones. The ecosystem of the floating islands needs to be able to survive drastic climatic changes. They need to be able to strive with hot, wet, equatorial weather, as well as with dry sunny weather, or frosty conditions, changing on a timescale of years.
How you adapt your flora and fauna to this, is up to you. I wouldn't think, its impossible to do, but it requires some thought to come up with a sketch of an ecosystem that can tolerate such drastic climate changes. Maybe some species will drastically change their appearance with the changing climate, while some others will become extinct outside of their home climate and only survive in the form of spores. This could lead to quite some interesting dynamics, imho.