Why does PCB copper surface oxidize when I apply flux on its surface?
If you look at the datasheet, you'll see a few important tips:


This flux must be cleaned after you apply it, since its corrosive, and shouldn't be used on bits that you're not soldering. The latter is true for most (all?) fluxes. The reason that water solubility is important for this flux is so that you can clean the pcb with water, and dont have to use other chemicals to clean it. If you dont want to have to clean the flux, you could try no-clean flux, which is less corrosive and doesnt absolutely need to be removed after soldering (although you still should remove it). In order to remove this kind of flux, you will typically need chemicals (like isopropyl alcohol).
I'm not sure theres a way to fix the damage already done to the board, but you could try cleaning the corrosion off with alcohol and scrubbing and then protecting the board with some type of conformal coating.