Layering and wicking.
Back to layering and wicking in a minute, but first note that 3,000 meters (9,842 feet) isn't really all that high compared to what you are aiming for, which is above 5,500 meters (18,045 feet). That extra 8,000 plus feet means a temperature drop of 16 degrees F (9 degrees C) to 24 degrees F (13 degrees C), more if you go higher. There may also be stronger winds higher up. This will be at least partially (maybe completely) offset by the lack of shade, higher insolation from above, and greater reflection from below and the sides if there is snow.
The extreme variation in temperature that you will feel, sometimes in a matter of minutes, from shifting cloud cover and moving cloud patterns and varying wind patterns makes layering absolutely essential.
From Chapter Five in Doctor on Everest by Kenneth Kamler, MD on climbing in the Western Cwm (6,000 meters to 6,800 meters):
The sun came out, and I stopped to remove my jacket and balaclava, but
by the time I got them off and into my pack, the sky clouded over and
I had to take them out again.
Earlier in the chapter, Kamler says:
It was getting hot. I took off a layer of clothes and switched from
my wool hat to my wide brimmed sun hat.
A bit later, Kamler and his companions faced heat exhaustion:
Our last precaution [after we each drank a liter of water] was to
strip down to a single layer of polypropylene ...Sustained exertion
could cause dangerous heat buildup, so we'd finish the climb in our
The next day, Kamler was out on an acclimatization hike
despite a freezing wind and wondering where the Cwm was that I had
climbed yesterday in my underwear.
You can face the same variations at lesser intensity at much lower elevations, so layering is always a good strategy.
As for wicking, it sounds as though you need better wicking technology in your base layer, or wicking in your next layer too.
Wrapping a towel or bandanna soaked in icy water around your neck in sustained hot conditions will help, and I have read an account of a climber who stuffed snow under his cap.