If they're dribbling, they could lose prime on the injector line in question, but not the whole fuel system. Look for an air leak somewhere allowing the system to bleed down. The lift pump should have a check valve on the supply side and the overflow valve should form a check valve on the return side. Be aware that the fuel preheater thermostat is a very common source of air ingress. If the fuel system gets air in it, it will all collect into a big bubble once the engine turns off and sits.

The 603's have a plastic disc in the overflow valve that warps and prevents a good seal. After fighting with that in mine, I swapped in the style from the 617 and have had 0 issues since.

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Current stable:
1995 E320 149K (Nancy)
1983 500SL 120K (SLoL)

Black Sheep:
1985 524TD 167K (TotalDumpster™)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)