To shreds you say,
I see now that's why you would need a freezing biome to dig down through the ocean and then put the aquifer drain
No you don't.... You also don't need an aquifer. Carving a fortification in the map edge creates a perfectly good infinite water drain. One of the things which should be mentioned about aquifers and map edges is that they are infinite water drains - not only do they accept unlimited water without every becoming full (it's best to say the water vanishes from existence), they accept water at an unlimited rate also - a single aquifer tile can absorb the whole ocean in a single tick. Because pressurized water (that is, water falling into water from above) happily teleports through 7/7 water, you also don't need wide drainage shafts, 1x1 is perfectly serviceable, and in some ways superior to wider shafts (it improves the ability of water to teleport).
Anyway if you want to use a non-freezing map, the trick is to dig out the complete drainage system below the ocean floor. Now destroy the floor of the ocean using a cave-in (collapse some constructed floors). It is also possible to use (possibly sacrificial) miners to dig downstairs into the ocean floor from below. I say "possibly sacrificial", I have done it without losing miners, but where exactly they'll get flushed to is extremely unpredictable. If you put many escape stairways (that is, an upstairs through a diagonal to cut off water pressure) then they will probably be able to escape before drowning. Also if you use floodgates to temporarily block off most of the drainage system the destination of the miner will be more predictable, making it easier to provide an escape route. You could also embark with an adequate swimmer, who will under some cases have a better chance of surviving.