That looks like an OMC engine, likely the 18 hp with two opposed cylinders. If the cylinders have the spark plugs located below center of each cylinder, it is the 18 hp version. If the spark plugs are located above center of the cylinders, it is the newer 22 hp version. The only significant difference between the two models is the cylinders, with the 22 hp version being a little higher compression.
My best guess is that your carburetor is in need of a thorough cleaning and rebuild. When you do this, make certain to replace the float as well as the needle valve that it controls. The float is not part of any carb kit. You have to order it separately. You also need to clean the accelerator pump passages, which don't clean easily with just a soaking of the carb body. I have used a single strand of some 20 gauge stranded hook-up wire to push the gunk out of mine. A new gasket and accelerator pump diaphram comes in the kit.
The mechanical fuel pumps on these engines are pretty much unobtainium (no longer available) and if you do find one it's been sitting on a shelf for years and the rubber parts are dried and cracked. I bought an automotive electric fuel pump and installed it in my Truckster. It needs to be low at about the same height as the tank outlet. It needs a fuel filter between it's inlet and the outlet of the fuel tank. You can then connect it's outlet directly to the inlet of the carburetor. These pumps self regulate the fuel pressure, so there is no need for the 3 port bypass filter any longer that bypassed the excess fuel pressure of the old fuel pump back to the fuel tank. Get rid of it. Before connecting the hose to the inlet of the carburetor, check to see if there is a tiny strainer/filter in the carburetor inlet. If there is, use a pair of needle nose pliers to pull it out and discard it. There is no replacement available, and by now the original is trash so just discard it. You won't need it anyway. You can just leave the old fuel pump on the engine, but not hooked up, or make a plate and gasket to fit the hole and bolt it in place. Leaving the hole open will allow dirt to get into the crankcase, so you either need to leave the old pump there or replace it with a DIY cover and gasket. Check the motor timing too, with a 12 volt light bulb. See the manual for how to do this.
The photo of the carburetor looks like a Keihin. You should find the name on the body of the carburetor somewhere. It's a Japanese carburetor and they are quite good when working properly. They also used a similar design Rochester carburetor for some of these engines. Cleaning and rebuilding the carburetor and replacing the fuel pump on my Truckster made a huge difference in how well my Truckster motor runs. When starting, I just switch the ignition on and wait a few seconds for the fuel pump to build pressure, and then turn the key further to engage the starter. Manual choking is required when the engine is started the first time each day, but the rest of the day it will start without choking in about 1 revolution. Doing these two things made a new Truckster for me in the way of easy starting and performance.
Keep the fins on the cylinders and crank case clean and the damper doors that are controlled by the bellows temperature sensors working or the engine will overheat. You really only need these doors and bellows assemblies if trying to get cab heat from the exhaust, so I don't know if your golf cart even has these. The motors will work better without these, if you have any problems with overheating. I can tell you how to leave them in place but block them from use easily if you have problems. Mine was done 5 years ago.
A manual is available in Acrobat pdf format on the
www.sillylittlecars website. From the main screen click on "Manuals" and then "shop manuals" in the text on the next screen.
These manuals don't cover most of your cart, but there is a section in the "supplement" manual that covers these engines very well. OMC (the outboard motor company) once owned Cushman and designed these engines to use in the Cushman vehicles. There were 3 different versions, with almost all of the parts the same for each one. I told you about the 18 and 22 hp versions, but there was also an 11 hp single cylinder version, also with almost all of the same parts. They just blocked the hole where one cylinder would be in the larger engines. These single cylinder OMC engines were used in a model of motor scooter that Cushman had made.
The rubber T connection behind the carburetor needs to be examined for leaks too. Be careful not to damage it because it is likely very brittle from age. These were available last time that I checked, but not cheap.
www.directparts.com will likely be your best source for motor parts or most drive train and steering parts. Sheet metal body and chassis parts will likely only be found from a wrecked unit of the same model, or made yourself.
I check in here most every morning. If you have any other questions, just reply to this post and I'll try to help.
Charley