Pumb only pumps a small amount out of the basin. Float needs adjusting to go all the way down and empty the basin. How do you adjust the float/bulb to travel farther down without shutting off after only a short distance?

probably not adjustment but something under the needle that is not letting it seal off properly
take the carby off , remove the bowl, tip upside down , pull the pivot pin out and remove the float carefully
there should be a tab which is slotted onto the end of the needle ( possible a small spring in the end of the needle as well)
remove the needle and check the rubber tip for damage /wear
unscrew the seat and remove and dirt , hair, grass that is preventing the needle from sealing off properly
at this point you can either replace the needle and seat or go to the mower shop and get a new assembly if it looks faulty
float levels seldom need adjusting as they are set correctly from new unless the float itself is failed

You need to adjust the amount of free swing the float has between the start and the off postion. With the pit empty the float should be hanging down low approx` 2" off the bottom of the tank and the start postion should be approx.` 2" from the top off the pump. You can use cable ties to tie the cable to the pump body but often there is a clamp arrangement. it`s trial and error until you get it right.

The problem is with the float switch. I have attached the manual. There is probably something keeping the float from falling back down. Check you sump basin and make sure nothing (including the power cord) is interfering with float.

Float lever out of adjustment on fuel pump module assembly. Remove assembly, float needs to be adjusted up so that "low light" is triggered before you get to empty.

Inside the tank, on a lever arm there should be a cylindrical float that rises and lowers with the water level. Attached to the float is a float rod with a clip that can slide up and down the rod. You can adjust the water level by sliding the clip down to reduce the amount of water entering the tank. As the water fills the tank, the float rises, pushing up the float rod, which in turn pushes up the level that controls the valve allowing water to enter the toilet.

not on the float, but on the fill tube cap,, its a small screw that you can adjust the arm drop with the float on the end of it, If its adjusted all the way and is still filling to much,, just bend the bar holding the float down

Yes it is suppose to fill tank nearly full then pump down and shut off. I would check to make sure that the float switch is not hanging up on something or perhaps rubbing on the tank wall that is stopping the pump from running long enough to empty tank.
Check the placement of pump and float to make sure they are not hitting on anything. Adjust float as needed to allow for max pump down. Also make sure that there is a check valve installed in discharge line so that any discharge can not run back into tank when pump shuts off. Hope this helps you. Thanks

what happens is the float is not staying open for long enough, by shorting the chain that goes to the flapper might help but you can only shorten it so much or the flapper does not close all the way then the toilet is just running and running. No you don't want that so the other thing you can do that might work is when you go to flush the toilet just hold the handel down for 5 seconds this will keep the float up and open for the Right amount of time. Other wise the other way to fix it witch is a old school trick is that you need to find a nice small piece of foam, witch floats and is about the size of a old half dollar and as least half inch thick, you place that on the chain that goes to the flapper so when the toilet is flushed the flapper will float and stay open . IF you do this just make sure to use a good piece of foam and secure it properly where it wont come off the chain.

The fill valve probably has a float that surrounds the pole that extends from the bottom of the tank. The float usually has a metal rod that comes down with a clip to adjust the float. Move that clip up so that the float sits higher-this should add more water to the tank.

Carburetors have a Float Bowl on the bottom. This float bowl has a small amount of fuel, that the carburetor uses. (The float bowl's amount of fuel, is replenished from the gas tank. The fuel is gravity fed from the gas tank, through the Fuel Line)

The float bowl has a Float Needle inside. The float needle is attached to the Float. The Float resembles a thick doughnut. On one end of this doughnut, is a flat piece of metal shaped like a U on the end. Looking at this -> U shape, there are two holes at the top. There is a metal pin that goes through these holes, and the pin is attached on it's end's, by the main body of the carburetor.

The flat metal U shape, and the metal pin, works together to form a type of hinge, allowing the float to pivot on it.
Where the U shaped hinge is attached, the float needle, clips to it.

The float needle has a tapered neoprene tip, and sits in a Float Needle Seat. This seat is a barrel shaped piece of brass, and has a tapered hole at the bottom. Fuel comes up through this seat, and the float needle regulates how much fuel can come in.

As the float drops down, it draws the float needle out of the float needle seat, and fuel comes in. As the float bowl fills with fuel, the float rises, and pushes the float back up, also pushing the float needle back into the float needle seat, cutting off the flow of fuel.

There is a specified height of the float, in relation to where the float sits in the float bowl. If the float sits too high, the float bowl fills up with fuel, and overflows. This overflow goes through an Overflow Tube, on the carburetor.
(If the float sits too low, the engine starves for gas)

A LOT of times, crud builds up in the gas tank, makes it's way through the fuel line, and gets into the float bowl. It can get in-between the float needle, and the float seat, thereby flooding the carburetor out, and fuel comes out of the overflow tube.

The float bowl is also attached to the carburetor with a brass main jet nut. At the bottom of the float bowl, this will look like a brass head of a bolt. Under the brass head is a small O-ring. This can leak. First step is too insure this is tight. (Do Not Over tighten! This is part of the main jet, and is expensive! It's Brass)

Where the outer part of the float bowl itself, attaches to the carburetor, there is another O-ring.
A large one. This O-ring, like the main jet O-ring, can deteriorate over time. Additives that gasoline has in it, heat from the engine, and heat from the sun does this. Plus just age. Neoprene breaks down after time.

It could also be that the neoprene tip of the float needle has broken down.

You indicated you adjusted screws on the bottom.
These are the Main Jet, and Idle Jet -> Air Mixture screws. All they adjust is how much air goes through these jets.
The amount of fuel going through, is predetermined by specific passageways, that are drilled through the main body of the carburetor.

This only comes into play when the engine is running. With the engine off, and fuel leaking out, you have crud under the float needle, or one or more of those O-rings has deteriorated.