Stinson Pilot reported an engine fuel starvation incident due to a stuck fuel selector valve; resulting in a forced landing at a diversionary airport.

Narrative

The following is a summary of a cross country flight in a rented Stinson; during which we experienced fuel starvation due in part to a stuck fuel selector handle; and made an engine-out landing at ZZZ. I had recently been checked out to fly the Stinson; and this was my first cross-country flight in the model. I had around three hours of experience in the plane prior to the incident flight. Lack of appropriate checklists in the airplane and my own lack of experience with the plane's legacy fuel system were contributing factors.We took off from ZZZ1 around XA:30. I had tested fuel from all three drains; then topped off both tanks while noting the (uncalibrated) fuel dipstick levels before and after. During the run-up; I used the fuel selector switch to confirm full readings on both tanks. Critically; the fuel selector valve was not tested. It would turn out to be stuck in the left-tank position. The minimal checklist in the plane only contains the items 'fuel gauges' and 'fuel on'; both in the preflight section. I had gotten in the habit of supplementing the Stinson checklists with my own checklists for another aircraft; but of course the other aircraft has two fuel gauges and no selector. Before takeoff; I had requested flight following to ZZZ; and had filed a VFR flight plan for the same. I planned to either land at ZZZ as a fuel stop or; if I determined that we had sufficient fuel; continue to our destination of ZZZ2 and make the fuel stop on the return journey instead. For that reason; I kept the fuel selector and gauge switch on the left tank for most of the flight; I wanted to know that I had at least twice as much fuel as indicated. I planned to switch tanks when the gauge read 1/4 full; as necessary. I was not conscious of the CG concerns of unbalanced tanks. We reached the 1/4 mark before we got to a visual waypoint; and I decided to switch tanks before we would descend to check it out. However; at that point I found that the selector handle was immovable. I twisted it with as much force as I thought I could without damaging it; before concluding incorrectly that the handle was inoperative; and that the switch superseded it; serving as both gauge and valve control. I believe the large number of inoperative components on the dashboard added to this perception; but so was lack of familiarity with the aircraft systems. I had read and highlighted the entire POH when I did my previous Stinson checkout flight over years ago; but my most recent flight was performed entirely in the pattern; and we never touched the fuel selector. I had also reviewed the POH prior to the incident flight; but the difference between the gauge selector and the valve control never came to my attention. Both components are quite hard to see in the grainy POH image.After I had satisfied myself that I was now on the full tank; we descended briefly to check out the visual waypoint; then began a cruise climb to 6500' MSL. The last leg of the flight; if I did choose to continue past ZZZ; would be over the coast; without a highway as an emergency landing option -- so I wanted to have plenty of altitude available. As we got closer to ZZZ; I observed that the right tank was still full; or nearly full; and decided to continue to ZZZ2. At this point; the engine suddenly lost power; and the plane pitched forward. I immediately trimmed for best glide; noted that we had plenty of altitude to reach ZZZ without engine power; and began troubleshooting. Electrical power was uninterrupted; and the propeller was either windmilling or turning under partial power. Adjusting the throttle initially had no effect; but I thought I noticed an RPM drop from about 1000 to less than 1000 when I pulled the throttle all the way to idle. I switched the fuel gauge selector from right to left. It struck me that the reading was now below 1/4; but it was not zero; and it was oscillating. My guess at the time was that the lower reading was related to the different pitch of the airplane; and I still did not realize that additional fuel had been drawn from that tank. I switched back to the full tank on the gauge and then tried turning the valve handle with even more force than previously; but it would not budge. I applied carb heat; then went through my aircraft checklist; as there was none for the Stinson. This included an attempted restart of the engine.When none of this was successful; I made a call on ZZZ CTAF advising of my position and intentions; and alerting any other traffic in the area that this would be an engine-out landing; and that I needed landing priority. There was no traffic in the pattern; but there was a Cessna who would be landing after me. I made another call on short final. I did not attempt to cut off the fuel; as I expected an uneventful dead-stick landing; and there was still a chance that the airplane was operating under partial power. I waited until I had the runway made; then applied one notch of flaps; and landed normally. The propeller windmilled briefly; then stopped completely. I had enough momentum to pull off onto a taxiway; but not enough to completely clear the runway. I made another call to advise the landing traffic behind me that I would need to get out and push the plane before the runway would be clear. As I did so; the airport attendant arrived in a car and helped me. I made one more call after the plane was clear of the active runway; then turned off the avionics and master switch. We pushed the plane all the way to the ramp; where I tied it down and began investigating the problem from the safety of the ground.On a call with one of the airplane owners; I still was not able to turn the valve handle. However; now that it was clear that the stuck handle was the problem; I used both hands and I did get it unstuck. By the time the other owner called me a few minutes later; the valve handle was moving freely; and even my wife could turn it. After satisfying myself that I understood what had happened; I filled the left tank and did a thorough preflight with special attention given to the fuel system. We then departed for ZZZ2 and arrived without incident. The following week; I returned to the airplane after it had sat idle for several days; and again was unable to budge the fuel selector handle. The owners agreed to fix the handle and put better checklists in the airplane; and I alerted the CFI who had done my checkout to the need for specific emphasis on the fuel system; the selector works differently than that of the mainstream training airplanes on the flight line; making a mechanical issue more confusing and more dangerous.