Transmission not shifting smoothly

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A drain and fill would just be the 3ish quarts from the pan out of, what, 12 or 16 quarts in the entire system? Assuming I correctly check the level and have the correct amount how is it possible to feel a difference with only a fraction of new fluid?

As what others have said replacing too much old dark fluid can cause the transmission to slip. The reason I and some others suggest just doing a drain and fill (3.5-4 quarts) so it changes only about 1/3 to see how the transmission reacts. If it’s ok, then you can likely do a flush.