Parts interchangeability for your tractor is determined the following way.
Find the ID tag on your tractor's dash tower and carefully record the serial number.
Then go to our Technical Library, click on the PARTS MANUAL section and find the Kohler powered tractors sub-section. Once you open that up, scroll down and find the parts PDF that coincides with your serial number.
If you are seeking major parts for your tractor, then ask the person selling the parts to give you the serial number of the tractor those parts were stripped off of.
Once you know that, you can go back to the Tech Library and find the parts manual that is correct for that donor tractor.
Compare part numbers.
The 444 was made from 1969 to 1988 and underwent many, many changes in the way it was built during that period. Do not assume that any part will interchange just because the hood has a 444 decal on it. Many of the "wear parts" can be bought brand new from local bearing supply houses as can some other parts.
As for the engine, I urge you to take this job on yourself unless you have deep pockets. If you have reasonable mechanical skills as well as regular hand tools, then you should pull the engine and tear it down. When you get to the bare block stage, take the block, crank, valves, valve springs and camshaft into an experienced machine shop that knows Kohler K engines inside out. Let them bore the cylinder, deck the block, mill the head flat, turn the crank, do the valve job, profile the camshaft and re-assemble the basic engine. You can bring that home, put the flywheel, engine tins etc back on and then bolt the engine back into the tractor. That's how you save money and still come away with a professional rebuild.
Now.....let me ask you this. Did you ever get this kind of advice on MTF? :sidelaugh: