If an engine under load misfires or stumbles, the problem likely involves the power valve inside the carburetor. In cases of rough idling, the issue may be a leak present between the carburetor and the manifold. Hesitation or stumbling when accelerating commonly results from the condition of the carburetor itself.
If the engine is under load (in neutral without throttling) and misfires, the power valve inside the carburetor may have a failing diaphragm or be clogged with dirt or fuel varnish deposits. To rectify, you need to replace the valve. This is typically inexpensive, as a backup valve is commonly included in carburetor rebuild kits.
If the vehicle is flooding, it is possible that dirt gets into the needle valve and is prohibiting it from closing. Additionally, the float inside the carburetor's fuel bowl may be leaking or set too high. Repairs of this kind typically do not require a full carburetor rebuild, but the purchase of a rebuild kit containing the proper gaskets.
If the vehicle stumbles during acceleration, the carburetor may be dirty or misadjusted. In such cases, the carburetor may also possess either worn throttle shafts or a weak accelerator pump. For example, the accelerator pump injects an extra dose of fuel into the carburetor's throat when throttling, a process that may be compromised if its diaphragm or piston is damaged or ineffectual. For these particular problems, you may need to either rebuild the carburetor or replace it outright.