While it is a shame that you did not get an answer here, had you read other threads on this forum (the search function is wonderful if you know for what you are searching) you might have known to look at the overflow valve and perhaps for an air leak between the lift pump and the tank (like at the strainer/preheater).
I have never heard of a shutoff solenoid being "weak". A solenoid is just an electromagnet with a moveable slug. In this case the electromagnet has two coils. The coils are either good or bad. The circuits that energize them can fail or the plunger (slug) can get dirty and stick in the bore. The rubber boot on one of mine got soft and gummy with age. It stuck to the plunger and jammed in the bore so that the pull coil was not strong enough to do its job. I removed the boot and cleaned the solenoid. It has worked well since. For what it is worth, the shop you went to may have seen an opportunity to sell you something you did not need. I hope you kept the old solenoid. From the symptoms you gave both the pull and the hold coils and energizing circuits were working well.
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96 3500 4WD, 47RE, gauges; 97 3500 4WD 47RE, gauges; 97 3500 2WD 5-spd; all club cab. 98.5 2500 2WD SB, 5-spd, gauges, quad cab