Hi!
A while ago we replaced the flush valve in our toilet. It no longer runs at random times, but in order to get a proper flush you have to hold the handle down or the flapper closes too soon.

The water level is correct.
If you hold the handle down, keeping the flapper up a bit longer, it flushes well.
When you flush, the flapper DOES open all the way.
It's almost as if it is just flushing too fast.

I have adjusted the chain length both by changing where the hook is on the links (changing length of the chain) and also by hooking it to the different holes on the lift rod. (It is a plastic lift rod, so I can't tweak it like in the old days.)

I haven't messed with the blue foam floater at all because I have no idea why it is there and what it does.

Thanks for any help.

You need to do some adjustment; with the flapper seated the linkage wants to have 1/4" to 1/2” of slack. Too much play and the flapper don't get pulled back enough giving you a short flush. Too tight and it lets water seep past the flapper, adjust the chain by the link and if it needs fine tuning, bend the flush lever rod a bit until you have the desired play in the linkage, for info see the image below, also check the water level in your tank should be 3/8 to 1/2" below the top of the overflow tube when the tank is filled, and the little tube to be connected to the white overflow tube.

The angle of the flapper also has a slight amount of influence.

With the play, the flapper will stay up and not seal right away.

Move the blue foam floater down the chain.

Ok, I moved the blue floater down. (I noticed it was down further on another toilet in our home.) I will toy with the chain some more. I will try bending the plastic arm. I think our water level is high, if anything. It fills all the way to the top of the tube and only when it overflows does it stop.

Oh and thanks everyone. I'll let you all know if this works!

If your fill valve lift arm is plastic you can't bend plastic, in that case it may have a screw on top of the fill valve to adjust the water level, in your case the water level is high and you do need to adjust it, see the image below, if your doesn't look like this please post a picture, Thanks

By blue floater, do you mean the float ball like the red ball (#1) above or do you mean a float on the flush chain.