That white valve is the transmission vacuum valve. Probably doesn't have anything to do with what you're experiencing, and neither does the EGR or vacuum system for that matter. If the car goes through the gears alright then it's not the issue. You could check it with a Mityvac or similar to be sure but I don't think your power and engine function would be affected by a vacuum problem. Most likely you have cracked rubber and/or plastic fuel lines that are giving way and allowing air into the fuel system, or the strainer in the fuel tank is clogged wth fungus or gunk accumulated throughout its life which is only now being loosened by the new fuel being bought and sold. There may be a possibility of the lift pump which is mounted to the side of the injection pump getting weak. Definitely a fuel delivery problem however, and fairly common as I am experiencing somewhat similar issues, but my car hasn't stalled out yet. Probably just a restriction in the fuel system somewhere.

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'02 BMW 325i
'85 300D 450k
'93 190E 2.6 170k(killed by tree)
'08 Ducati Hypermotard 1100S 6k
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