Okay... here's the latest.
I bought a new breather, part number 31337B. Here's a pic of the parts:
I installed that. Then I rebuilt the original carburetor and installed that. Then I put the original muffler back on.
It started relatively easy, and I was able to make carburetor adjustments right away. I got it where I thought it was okay... adjusting the bottom screw approximately halfway between where it wanted to stall. Adjusting the idle screw down from a heavy rev.
After about 15 minutes of this, it stalled. I figured it was due to the fact that I was still adjusting the carburetor at the time. But it took several pulls to get it to start again, and wouldn't stay running for more than a minute. Kept trying, wouldn't stay running for more than a minute now. Went to using starting fluid and finally it wouldn't run at all.
I pulled the spark plug and it looked dry. So I let it sit overnight.
I went out today and was able to get it started and it ran for only a few seconds and only when it had the choke on. No matter what I did it wouldn't run unless it was choked. I decided to change out the idle screw (I was using the original which looked pretty good but replaced it with the one from the carb kit) and readjust that a turn and a half out and not really any luck. After research on google, I figured the jets were plugged (though it was a new jet assembly and adjusting screw) so I put yet another one on (I had one left over from another carb kit). This time I used starting fluid to get it started, and it finally took off so where I could keep it running again. It would run for a few minutes, I could adjust the bottom screw for gas mix and the idle screw to where it was running okay again, and then it stalled out again. Again, tough time starting to the point where it wouldn't start at all.
All the while, the engine is dripping oil from what looks like to be the front and the back. Pics below, but from the back of the engine it looks to be crankcase breather again (including dripping from the new tube) and now it looks like it's leaking from the governor where it is bolted into the block.
It's a mystery where it's coming from in the front of the engine. I suppose it could be the head, but it could also be a crack in the block too.
So it could be that I installed the crankcase breather all wrong. It could also be that the governor rod is leaking, or it could be that I have a head issue or a cracked crankcase. Either way, I'm flummoxed to the point that I'd rather take it to someone who knows what they're doing or buy a replacement engine... assuming I can find another Tecumseh H35 because I still want to keep it close to original. I can't imagine how hard to find or expensive that would be.
I guess I'll start researching small engine mechanics who specialize in engine rebuilds and restores. I'll also start shopping around for replacement Tecumseh H35 engines. Bottom line is I've reached the end of my knowledge on the issue and I could keep stumbling around but maybe it's just time to call upon a professional to fix it right.
Here's some more photos...