Hi, I'm new to this forum as of last week.

I just picked up a used Toro 524, and am trying to get it ready for the snow forecast for later this week. I don't have alot of $, so I need to DIY.

After a good deal of priming, the engine turns over, and I can get it running in neutral. No matter how I adjust the main (?) jet on the carb (using the screw at the bottom and in the middle of the bowl), I can't get rid of a small, but discernible cough in the engine. The cough happens once every 5-10 turns of the engine. Right now, I have the carb's main (?) jet screw adjusted at 1 3/4 turns out from dead bottom (i.e. all the way in).

As soon as I put the blower in gear and into the snow, the engine stalls.

I put a new spark plug in, but same problem. I put 'sea foam' in the gas and in the air intake, and started it up again, but only a small improvement.

Anything else I can adjust to get power to the engine? I'm guessing the small cough I hear while the engine is running is part of the problem, but am not sure.

Thanks for any help.

p.s. Just went out, got a compression tester, and the compression is reading at over 120 psi.

Depending upon age, it could be any number of things.
If old unit, could be a intake/exhaust valve problem, carburetor restriction, ignition problem, water in the fuel, a bad spark plug, a blown head gasket....
How old is fuel? (it does degrade and go to crap)
Does choke help it?
Does it idle fine without stalling?
Does it run at full speed (with cough) without stalling?

There were a few issues so far from your list:

I really needed the blower for today's storm, so took the machine into the shop, put the repair on Visa. They found water and bad gas, and drained the tank, flushed the lines, and rebuilt the carb. Now the machine will take a medium load, but still cuts out, not giving full power.

It is an older machine, I think from the '80's, and heavy as a tank.

I changed the spark, but I checked and the new one got black pretty fast.

The choke does help it - now the engine won't even run properly unless the choke is 1/4 on. If I turn the choke off, the engine sputters and dies. Witn the choke 1/4 on, it still blows a bit of snow, but with more than a small load, it will die.

The engine idles ok without stalling (but the choke has to be 1/4 on), and if the load is minimal, it will run without stalling. I'm still having the issue where under more than a small load it will stall.

Before I took it to the shop, I checked the compression and it was over 120 psi.

Sounds like there is still a restriction in the jet of the carburator, take it back to the repair shop and make them fix it right.

sounds like the low idle screw needs to be taken out a little bit

I took the blower out this afternoon. I adjusted the main jet (in the bottom center of carb bowl) so that it's now just over 3 full turns out from being fully screwed in. The blower runs alot better, there is power, and the choke goes all the way in without a problem. But, after about 10 min's the engine cuts out, and there's like a muffled sound, even more when I give it gas. I'm guessing this means I pulled the main jet too far out, allowing too much gas into the mix??

Two questions for those with experience:

1. can I damage the engine if I pull the main jet out too far?

2. Am I likely to be right that I've just pulled the jet out too far, or is there some other factor I'm not considering here ?

I'm planning to take it back to the shop, but I don't have a pick-up and it's killing my back loading and unloading this heavy beast of a machine. Plus I like learning about how the machine works as long as I don't break it.

I'd bring it back to wherever you had the carb. rebuilt - I think there's still a restriction. Any reputable shop would re-adjust a carb. they just rebuilt at N/C. I offer this to customers when it's warmer out as engines require a richer mixture in the cold (air is more dense then) and you can get it close in the warmer temps. but not "dead on."
As for damaging the engine, not really.
Overly rich mixture can promote carbon buildup on valves, but you'd foul the spark plug long before that, which would be an indicator to correct the mixture.
You can't "pull the jet out too far" - the screw has many turns before it can come out, and more than 2 turns is usually at the edge of maxium richness.

I took this beast back to the repair shop, told them there was still a problem, and once I got it home, it basically had the same issue. They seem to adjust the carb without a rich enough mix, 'cause right out of the car, the blower cut out under a heavy load and I had to re-adjust the carb, pulling it out about half a turn more. Why they would adjust it wrong is beyond me. Once again I got it to work under a heavy load, but it still cuts out randomly.

One thing I neglected to mention before: when I pump the primer valve more than 3 or 4 times, gas starts to leak from the carb. Is this normal? Apparently the shop pulled the carb apart for the 2nd time and cleaned it again, so I can't imagine why any gaskets would be failing now. Anyone have anything else to suggest? Is it normal for the carb to leak gas when I pump the primer bulb alot ?

Thanks for everything so far!

Yes, you may be pumping primer too many times. When you pump the primer you are pushing air into carb bowl chamber, with too many pumps and there is no place for fuel to go but out the front of carb.
When you picked up your snowblower from the SE shop did you ask them to blow some snow to make sure it worked good or just take their word for it? I also test my customers stuff before they come and then again when they come to p/u unit.HTH

If this unit is from the early 80's, it may have point-based ignition and may need them replaced along with the condensor. They switched over somewhere in there - don't remember exactly when.
But, this would account for the blackened spark plug and not providing full power.

You could test the spark with an inline tester, while it's running, or use a load type tester.

The first digit of the Toro serial number is the year of mfg.

Write down the numbers, go to:
https://lookup3.toro.com/partdex/index.cfm?xCaller=Toro
You'll need the serial number too!
Look at the engine breakdown, if there's a magneto shown behind the flywheel, it has points.