Hi Bampter, and welcome to Outdoorking.

The usual cause of hunting is lean mixture, as you obviously know, since you have been focusing on the fuel supply. Incidentally I hope you cleaned and inspected the fuel filter. However there is another, less common cause: friction in the governor mechanism, caused by, for example, a bent link which jams slightly, something rubbing against a lever, or a pivot that needs to be lubricated.

I suggest you look at the governor mechanism to check that it moves freely, but lean mixture remains the more likely problem. You should be able to verify this though, by restricting the intake air slightly when the engine is hunting, to see whether that clears it up. Most (not all) engines, including yours, have a choke which can be closed a very small amount to make this test. If your engine had only a primer, not a choke, for cold starting you could either obstruct the air filter's intake passage, or remove the air filter and obstruct the carburetor air intake. I do not recommend running engines with the air filter removed, especially if they are running lean, because they may "spit" burning fuel from the air intake, and this is hazardous.

Here is the workshop manual for your engine:
http://golftechs.us/Manuals/Suzuki.pdf

There is a likely cause for lean mixture other than the fuel system: an intake air leak, on the suction (i.e. engine) side of the carburetor. This usually happens in one of two ways: the gasket between carburetor and engine is loose or damaged; or air is leaking directly into the crankcase, perhaps through one or both of the crankshaft seals. In most cases if these are leaking there will be oil around them.

Please try closing the choke slightly to establish definitely whether the problem is lean mixture. If it is, investigate the items I have listed. Either way, let us know what you find, and we can continue the diagnostic process if necessary.