TOTO CT746 won't stop running

[Lindag]

I have a Toto Drake CT746CUG#01 toilet that won't stop running. After I flush it, water runs inside the bowl while the tank is trying to fill up. If I stop the filler (black tube) by that white tube in the middle of the tank, the water stops flowing in the bowl and the tank finishes filling. Sometimes stops running. Sometimes I have adjust the float level (the little screw inside the green cover on the left) to get it to quit. I have never fixed a toilet before so I really don't know where to start or what tools I need. I can't find a video "how-to" for the toilet.

Is a CT746 the same as a CST746?

I have included a picture of the label inside the tank. I think it's the tank part number plus a list of parts used inside it.

We have a private well. We live in farmland so our water has sediment.
I have been looking on the forum and other places and I can't find any information about how to fix the problem. The manual isn't that great.
Right now the water is turned off.

Any help is greatly appreciated.

This is the tag inside the tank. It has the part numbers on it.

 

[Terry]

I wonder if the red line from the handle is sticking on the fill valve float. If the fill valve has dirt in it, it may also be a problem.
Pretty much any brand of fill valve will work with that tank. Korky and Fluidmaster are the most common.
You have the 746 series of the Drake.

 

[Sponsor]

 

[Breplum]

I recommend, first, to just get rid of that abominable "Toto" green fill valve (aka ballcock).
If it will fit, a good choice is the adjustable refill https://www.amazon.com/Fluidmaster-...hvlocphy=&hvtargid=pla-4584207581818721&psc=1

That Fluidmaster 400 valve has one $2. gasket that will fix it for a lifetime, should the need arise. That green abomination from Toto has no parts nor hope.

 

[Lindag]

Thanks a bunch!
I found this video. Now it makes sense.
Toilet Valve that Keeps Running:

 

[Lindag]

Thanks Breplum. I didn't catch your suggestion until after I started looking at videos. If the problem is beyond cleaning, I'll definitely be buying that part.