Just to update everyone. After getting my pump and filter in, the car is a lot smoother between gears. When rpm drops for clutching and then accelerating, the whole thing seems a lot smoother- no herky-jerky. I thought I maybe had a VANOS problem as well, but everything seems pretty good now.
I hear the smoothness might come from replacing the filter more than the pump, but my pump was super-loud before it finally died, so I'm sure it was not doing great.
Yay for new pump and filter.
2002 330Ci M Sport, 5-speed Manual
2013 528xi Premium, 8-speed Steptronic