"I Recently inherited a BobCat Snow Blower from the 60's or 70's. It has a Lausen 3.5hp (HR35P-2403P). I just installed a Tecumseh Rebuild Kit#631893. The unit runs way too rich, I run out of gas in 10 minutes, the muffler starts to glow red. I cannot get it to idle. I installed a new needle, seat/jet and diaphragm. During the rebuild I did not remove the welch plugs. I tried swapping the idle screw with the high speed screw (not sure which is which). One screw has a smooth taper while the other has a taper with a step to it (no evidence of scoring). Could I be missing a key piece that regulated fuel flow? The rebuild kit did not come with directions, so I just installed everything in reverse."
The idle mixture screw is the one with the step.
I assume you have any choke off, throttle connected properly with spring return, etc.
What happens if you close both idle and main jets? Does it starve? I would expect that you should be able to stop fuel flow totally. If this is not possible, your needles or something else is incorrect/worn or fuel is somehow bypassing the jets which is also wrong.
Did you compare the old and new parts to make sure they gave you the correct kit?
It also recommend removing the Welch plug and blowing compressed air through the passages to clean.
It might also be a problem in the diaphragm spring pressure but without seeing it, no way of knowing. The diaphragm acts against atmospheric pressure. There is a spring on the inlet needle which if missing would run very rich. Chilten has a detailed diagram - really no way of knowing if your assembly was done correctly. Also, warns again using harsh cleaners on non-metallic parts and clearing all vent holes.
However, I rather suspect that comment about running rich is not correct as an engine running very rich would lack power if it continued to run at all. Your throttle may be stuck wide open and it may be over revving.
Your public library should have some Chilten or other books like those listed in the section: References. These should include diagrams of the diaphragm type carburetor.