Hi, I have the same forklift and what WAS on it was an IMPCO Model K. Why, I am not sure, but this model is VERY hard to find. I replaced it with a Unisourse TSA/800115159. I purchased mine from ALTA Material Handling. 630-966-0450. $126.19

Ok, for my issue. The forklift was running fine until it ran out of fuel. Using the SAME TANK I had the tank refilled and put it back on the forklift. Opened the value and it started right up. Went to give it gas and as soon as you passed ideal, it started coughing - like it's not getting enough fuel. Turned off the tank valve and ran the forklift out of fuel again and disconnected the tank. Then reconnected the tank and just "cracked" the fuel open a little. I was thinking, well maybe the safety value engaged. It started right up, running smoothly, pushed on the gas pedal, and the same thing happened.
I nursed it out of the middle of the driveway trying to figure out what was wrong. Checked the "O-RING in the fuel tank to make sure it was still there. I spent the rest of the day on YouTube and the Internet trying to figure out the problem. I found one video that said, "PUSH THE LITTLE BUTTON ON THE SIDE OF THE VAPORIZER." That the button acted kind of like a choke. So, I did. Will shoot, I had FULL PEDAL AND IT WAS RUNNING GREAT! I released the button, and it went back to the same problem. More reach and I was convinced that somehow the vaporizer had gone bad. So, I replaced it with the one I mentioned above.
Surely, this would fix the problem - RIGHT? Nope, Started the forklift and I have the same problem. Ok, it's an old forklift so I thought, MAYBE the connector to the tank has worn out enough that it is not making good enough contact with the fuel tank to release enough gas when there is a demand. I ordered a new Connector, and O-Rings for the fuel tank. Put them on today, and the problem still exists.
THERE JUST ISN'T THAT MUCH TO A LP SYSTEM TO GO WRONG!
Last summer I put all new points, condenser, cap, rotor, plugs, and wires. Basic tune-up stuff. I checked the cap and rotor to see if they were still in good condition and they are. Again, all this started AFTER I RAN OUT OF FUEL AND PUT A REFILED TANK BACK ONTO IT!
The last thing in the line of possible components is the carburetor. So I am looking for a rebuild kit for that now.

Can anyone tell me what is going on?
I do have a video of the forklift running at idle and when I am trying to give it gas.

Thank you.

P.S Andrew, if I can figure out how to post a pic of the model K and the one I put on I will.