Avoid an aftermarket carburetor, if at all possible, 2 out of 3 are usually worse than what you already have and new OEM carburetors are expensive. I usually get parts from Part Tree or Jacks Small Engine. I am not a fan of Amazon, but I do use eBay pretty often.
Work slowly and take plenty of pictures for reference.
90% of the time, you just need to take the carburetor off and remove the fuel (anti-backfire) solenoid with a thin wrench and carefully tap the float bowl to remove it. Once removed, clean it good. Also check the solenoid for rust and sticking which is common. Clean the solenoid tip/pintle with carburetor spray and gently work it back and forth as you clean it.
Lay out a couple of paper towels inside of a soda crate box, then lay every part in an intentional pattern as you remove them. You don't have to take everything out or off the carburetor, just the important components.
Then remove the cross pin holding the plastic float and carefully remove the float. Watch how the needle is attached to the float because it will need to go back the same way. Clean the needle tip gently with a soft cloth and inspect it for any damage.
Next, you will need to inspect the seat area where the needle goes and use carburetor cleaner with a straw attachment to blow spray backwards from the seat area to the fuel inlet portion where the fuel line connects. This is the most common area for trash to get stuck and not allow the needle to fully seat.
If the needle seat is a tiny orange ring down inside the hole at the bottom, let us know, because those can be a real pain and you will need a special tool to replace it.
I have seldom had to order a carburetor kit unless something was really bad.
Assembly is just a reverse of disassembly. Be careful not to overtighten or force anything. Patience is your best approach.
Attach some digital pictures if you can.
Let us know how you progress on this new adventure.
Teryl Fixes All has some good small engine carburetor videos for all kinds, so check out some of his videos for a similar one to yours.