I went out and got going outdoors for a long time, so now I'm a bit tired. I went out and got going outdoors for a long time, so now I'm a bit tired.

I think "got going" sort of makes sense, but "was active" or "exercised" might be a little better. "Get going" is usually used when you start something, depart, or start talking enthusiastically about a topic.

It' feels kind of like heatstroke. It feels kind of like heatstroke.

Besides, my laundriesy completely dried thanks to the heat. Besides, my laundry completely dried thanks to the heat.

While "laundries" is technically correct, in general situations like this we usually just write "laundry." "Laundries" might be used if referring to specific groups or types of laundry. I would probably use, "At least" instead of "Besides," but it's okay as is.