Hmmm, it does sound like inorganic nutrients are quite available. Often in our tanks there is so much competition and herbivory that many nuisanse algae can have a tough time growing as prolifically as they would in nature given certain nutrient levels. Nitrate and phosphate concentrations that are the lowest detectable level on a hobbyist test kit, for instance, rarely occur in nature and usually only in areas of upwelling or severe pollution (raw sewage).
I would likely do a series of water changes, but check the TDS of your source water to ensure that you're starting with very pure water. I'd also check the nitrate and phosphate of a new batch of seawater, to make sure nothing is decomposing in the vat (sometimes just dust is enough) and influencing your input water. I'd also get your skimmer performing as well as you can--clean it daily for best results. I'd use a hefty amount of activated carbon. This carbon will remove many dissolved organic substances which can decompose into inorganic nutrients and fuel more algae. Lastly, I'd manually harvest algae from the tank as a means of nutrient export. Don't overfeed your fish during this time, but certainly don't starve them. Starving them (so often recommended) is a really terrible way to try to reduce nutrient levels, and isn't particularly effective anyway. It's better to do an extra 10% of the tank's volume water exchange than to starve ones fish IME.
Could we also see some new photos of the old corals and some photos of the new corals as well as perhaps a few tanks shots? A picture is worth 1000 words, often times.
Chris