Never had that issue myself. Acceleration while moving only seems strange if you manually downshift and 'punch it' and then the transmission tries to downshift again. You learn to let the transmission do the work in those cases or you learn not to really 'punch it' and ease down on the accelerator after YOU downshift to the proper gear. This is much smoother and you can still aggressively push the pedal down without forcing the transmission to try and downshift.

Sounds like you may need to do an accelerator reset.