Better use a plant than corals as a basis.
Corals are a very specific lifeform (they are animals, not considered being plants, but kind of a hybrid) that is adapted to
- stay put on one place (they have a "foot" which anchors them on a certain spot)
- Consist of living tissue and produce calcite of dead tissue which form the reefs after long times. Calcite is a mineral, porous and brittle, does not contain cellulose (the stuff wood is made of).
- live under water but close to the surface
- some can digest planckton or capture little fish etc. Most of them rely on photosynthesis though.
- are very vulnerable to changing environment conditions (water temperature, salinity, sunlight, availability of "food" etc.)
Floating corals could mean being captured in currents and have to deal with ever changing environmental conditions. I would argue that these creatures are not a good choice to start to genetically modify them into a essentially totally different species.
From the input given in the description I gather that you actually would like to have those floating islands which happen to provide wood.
As a starting point I would suggest to take a free-floating water plant (in symbiosis with a fungi for example) which either evolved naturally or was the product of some actual gene editing by humans - or some long forgotten experiment in a now submerged bio laboratory that grew and spread in the new water world.
Free-floating plants exist and are called "hydrophytes" (water plants) or "macrophytes" (to distinguish from not being algae). Here is an overview of those used as a pond plant (that is: sweet water). Wikipedia has some thin information about hydrophytes/macrophytes. I didn't check for some which would be able to thrive in salt water. But this is either much easier to be made plausible or maybe your world does not even consist of salt water in high concentrations maybe.
The needed changes would just be forming of a more stable version of the usually soft stems made of cellulose on cellular level. In fact they would form a bark. Maybe to protect against fish which eat them otherwise (also thorns or poison could be a thing) or to attract birds which feces would be excellent fertilizer for them. Their shadow would also attract fishes and other life to hide in and find protection from burning sunlight.
The bark can be elastic like willow rods (withy) and the islands are intertwined and could cover square miles if needed. They could also form the living ground for other animal species (insects) or plants which are even more tree like. Or the floating plant itself evolved a strong, tree-like trunk of which the pollen are ejected to reproduce. The characteristics could be as strong as a classical tree or more like bamboo for example.