When I first obtained my w124, I had problems modulating my accelerator as well. I have a manual, so the results were somewhat different (stalling, or too much clutch slipping).
In my case the solution was to thoroughly clean and lubricate all of the throttle linkage bits, including all of the balljoints, and plastic wheels, etc. I used brake cleaner and white lithium spray grease. No problems a year later.
Based upon past posts, I would say this is a common problem.
Hope that is your problem as well, since it's an easy fix.
1986 300E 5-Speed 240k mi.