Two things are happening. One is the cut in paint is drying before the rolled paint. You can't do one and come back to do the other later.
Actually you can do exactly that - cut in with a brush, let it dry, then roll. I learned to do it from experts on this forum.
I'd use flat/matte. It should not only help with the problem in the corners, it will reduce the noticeability of any minor flaws elsewhere with the wall surface.
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