My 1982 300SD did that till I found 2 bad Glow Plugs. On the SD the glow plugs stay energized even after the Glow Plug light goes off in the cab. About 30 more seconds. You can hear relay click after 30 seconds after the Glow Plug indication light extinguishes in the cabin. It may be the same as yours.

What I'm getting at is the glow plugs are there for starting the car of course, but also are there to stay on after then engine starts. If it starts ok once it is warmed up I'd look at the glow plugs. Easy to test with an Ohmeter.