603's will skip a bit on a cold morning that is normal. But it sounds like yours is skipping really badly.

It sounds almost like you might have a weak glow plug.

As to the being slow off the line I am not sure I have not had to deal with that problem yet. Maybe a turn of the ALDA would help.

Change the filters before you do anything though, a very dirty air filter will cause black smoke.

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1999 SL500
1969 280SE
2023 Ram 1500
2007 Tiara 3200