On April 19, 2019, about 1630 central daylight time, a Piper PA-28-235 airplane, N838JH, conducted a precautionary landing at the Dwight Airport (KDTG), Dwight, Illinois. The pilot and two passengers were not injured, and the airplane was substantially damaged during the landing. The airplane was registered to and operated by Aten, Inc, under the provisions of 14 Code of Federal Regulations Part 91 as personal flight. Visual meteorological conditions prevailed at the time. The pilot reported to the responding Federal Aviation Administration inspector, that he was in cruise flight at 6,500 ft. The engine started to run rough, so he applied carburetor heat, but the engine continued to run rough. He noticed the fuel pressure was zero, so he turned on the electric boost pump, but nothing changed. He then switched fuel tanks and the engine smoothed out a little with a slight increase in fuel pressure. The pilot then decided to divert to KDTG for a precautionary landing. A couple minutes later, the fuel pressure again went to zero; he switched to the left tip fuel tank, and the engine regained power momentarily. The scenario would repeat; after switching tanks, the engine momentarily regained power, so he kept switching tanks, it in order to reach KDTG. During the crosswind landing, the airplane drifted off the runway and the nose gear collapsed. Inspection of the airplane noted a collapsed nose landing gear and substantial damage to the engine mount. Fuel was found in 3 of the 4 fuel tanks. A mechanic and FAA inspector examined the airplane's fuel system. Fuel did not appear to be flowing properly to the carburetor. The fuel line was separated before the engine driven fuel pump and the electric boost pump activated. Adequate fuel flow was still not observed. Only a small amount of debris was found in the gascolator bowl. Both the engine driven pump and electric boost pump were removed from the airplane. An initial test of the engine driven pump found the it would not pump fuel. A restriction or any additional anomalies were not found with the airplane's fuel system. The engine driven pump and electric boost pump was then tested by the NTSB Investigator-in-Charge. When the engine driven pump was operated by hand, suction and pressure was felt on the pump's inlet and outlet ports. The inlet of the electric boost pump was connected to a hose and the inlet end submerged in a pail of water. A 12-volt battery was then connected to the pump; the pump ran but would not draw water from the pail, and only, "spit" a small quantity of liquid.