It will help a lot by just replacing the sector gear, but it will only be that much better if you replace both the gear and the worm gear shaft.
There should be a nut on top of the sector gear. I couldn't tell by your pictures if it had been removed. The splines on the gear and the shaft are tapered so it might take a little work getting it off.
You can also tighten up the steering my checking the worm gear shaft in the housing by the gages. A machine shop can build up the shaft and have bushings installed in the housing if needed. The U-joint shoulnd't have any play. You may have luck finding a new/used rear shaft with a tight joint or that one can be cut off and replaced. Same thing goes on that shaft where it passes through the steering post. You don't want a lot of play there.
Another thing that will affect the wheels wobbling is the tires. You need a matched set and make sure they have equal air pressure.
Some tractors had an aftermarket "wedge" installed between the lower bolster and where it bolted to the pivot shaft. This adjusted the angle of the lower bolster making it more vertical. This change makes the tractor easier to steer but I'm told you do sacrifice in that it may make the front it wobble.
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Today's Featured Article - The Cletrac General GG and the BF Avery A - A Bit of History - by Mike Ballash. This article is a summary of what I have gathered up from various sources on the Gletrac General GG and the B. F. Avery model A tractors. I am quite sure that most of it is accurate. The General GG was made by the Cleveland Tractor Company (Cletrac) of Cleveland, Ohio. Originally the company was called the Cleveland Motor Plow Company which began in 1912, then the Cleveland Tractor Company (1917) and finally Cletrac.
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