I also have asthma and prefer the cabin to be cool when possible.
It's been my experience that the cabin cools down better when the ship is not docked. It usually takes at least until 6 or 6:30pm to cool down on the first evening out.
From the deck plans I see that you're in a Veranda Cabin and from experience, I know that it does help to keep the curtains closed to the Veranda when you aren't sitting out on it or viewing it from the couch.
It also seems to help when we close the privacy curtains between the double bed and the couch. Since it was just my DW and I and we didn't use the sofa, I can't say what effect that had on the couch area, but it seemed cooler also.
My DW and I have resolved that we will buy a small electric fan that oscillates to help with the air flow on our next cruise. We are used to a ceiling fan at home. If you decide on a small portable fan too, you'll also need to bring an extension cord, there's only one outlet and it's on the left desk of the desk.
It was never warm enough that I felt I needed more medication than I use normally at home.
I just reread your original post and I would have to qualify my statements by saying we were comfortable with the temperature at 70 or 72 degrees.