Lawnmower wont start or runs rough

It sounds like you have a combination of hanging floats,and debris in the carb jets. The carbs will need to come off ,disassembled completely,and cleaned.Paying extra close attention to the jets. Remember,there are two jets in each carb,one for idle,the other is the main. Be sure to clean/soak them THOROUGHLY!! And it always helps on a bike this old,to add an inline fuel filter in case the tank has rust/debris inside of it. After reassembly,you will also need to synchronize the carbs as well.
Good Luck!

most likely your main jet in the carb is clogged, water in fuel, or a plugged fuel line, filter, or plugged in the tank where the fuel line connects. it is common to have a plug where the fuel line exits the fuel tank if there is a 90 degree turn in the fitting, debris may build up in the 90 degree area. check by replacing old fuel with new, then pull fuel line from carb and fuel should flow freely, look for a blocked fuel line if it does not flow. it it does flow, pinch off fuel line then remove the float bowl to drain remaining gas and give acess to the main jet. the main jet is located in the side or bottom of the lowest protruding area into the float bowl. run a thin wire like a guitar string thru the small jet hole to unplug, then squirt a little wd-40 thru jet hole to visually check flow. next, unpinch fuel line and fuel should flow from carb into float bowl area. then check to make sure the fuel stops flowing when the float is gently lifted to the top of its travel. if problem in this area, remove float pin, float, and needle valve and check for blockage in the needle valve seat. clean out with wd-40 and reassemble, add fresh gas and give it a try!

Running only on the prime is caused by the main jet in the carburetor being blocked. This can happen if you put dirty or old fuel in it or let it sit for a long time with fuel in it. Today's ethanolated gas goes bad quickly and absorbs tons of moisture which causes water to settle to bottom of the fuel tank, which in turn prevents the main jet from lifting fuel, it picks up water instead, thus causing it to die. So, check the tank for water, drain. Check the gas can, and get rid of the old stuff. If debris in either of them, you may have to take carb apart to clean the main jet.

The battery is only the beginning. The carbs are most likely gummed up. The idle jets are plugged, and the old gasoline in there wouldn't help. Your idle jet, (pilot jet, whichever name you prefer) is plugged up in your carburetors. If the bike was stored improperly, or old/dirty gas was in the tank, these jets get plugged up pretty easily. The idle jet is where your bike pulls gas while it has a closed throttle position. Its the smallest jet in the carbs and makes it hell to start a bike when they are plugged up. Good thing for you, is that you only have one carb, and it is very easy to take out and clean. The jets are under the BOTTOM cover of the carbs. You will see the main jet is on a Taller part of the carb and the idle jet is next to that. Here is a picture to help you, It isn't the exact carb, I borrowed this pic from another website. Number 1 is the pilot jet, 2 is the main jet and 3 is where the fuel comes into the carb. This is your float valve. This has to be clear of stuff too or it won't shut off properly and you will leak gas from the carbs. So get yourself a can of carb cleaner and start spraying!
Bruno

if this is a vertical crankshaft 3.5 then you may have the gasket and diaphragm inverted, the diaphragm must be against the tank, and the gasket against the carb for the fuel pump to work, also check the pickup tube screen for being plugged, water in the tank bowl or a film or varnish or oxidation on the main jet sock.

If this is a horizontal shaft, like a go-kart, all the stuff about plugged pickups and main nozzles applies, only big difference is if there was a lot of corrosion in the bowl under the carb, the main jet could be permanently plugged, also there should be a removable jet on the control lever side of the carb that will probably need cleaning as well, also when removing the carb from the tank make note of any fuel in the bowl, if there isn't any, make sure the pickup tube isn't plugged or split, the diaphragm spring is installed, and the bowl doesn't have a hole rotted through it.