Since you ran it out of gas it is very possible that some crud got sucked into the carb so a good cleaning will likely resolve the issue. Congrats and Good Luck
I went over the posts this morning and had an observation, the majority of experience here as lance also referred to is in some way or another on a track of diagnosing "Fuel Starvation". Propane was getting there too with hydro-lock, or lack of air to replace liquid volume through gravity. Carbs have a vent to atmosphere too.
I wonder what you might find if you get the tractor running full and observe the fuel filter, would it be full or mostly full at the point the motor dies? If you did the same thing but cut power while at full throttle and take the spark plug out immediately, would that plug be wet? If the motor is sputtering and you jiggle the filter would it have a noticeable effect? With the cap off? with the air filter off?
When or if you take that carb off you can spray lube such as wd40 or ? and if it has dye in it even better to check the passages and see if they aren't restricted or blocked. If you choose to rebuild, you might try resetting your high and low to factory settings.
If you haven't adjusted a carb before, factory setting will usually allow the motor to run, if it doesn't there are other problems.
lawdawg, in my own personal experience, if I ignore obvious signs and power through a problem at night, it usually ends up costing me the most money with less than favorable results....and yet I still find myself doing it, crazy right?
I think you found a good tractor, and after you work through the ignored problems from it's past will have a great tractor. Can't wait to see what it can do for you.
Cheers