Shortly after toilet flushes, you hear a tinkling sound for several seconds, as if water dripping inside. Sometimes also hear another sound like flushing even though toilet is done running,

Water is not draining completely from the toilet line to the sewage tank. If there is a 'low/little slope' section anywhere in that drain line, solids WILL settle and collect there and cause water to back-up in the toilet drain line. Then the toilet will not flush completely or at all until the 'soft blockage' is cleared out and the drain line emptied. Al 'gravity drain' toilets require a free draining/empty drain line that allows flush water to drain unrestricted. A basement toilet has a pump that 'forces' flush water up and out with enough pressure to sometimes force out the 'soft clog.'

There will never be a dripping sound from the bowl unless there is water on the floor indicating a cracked bowl or a gasket has lived its life and needs replacing . The flush kit could drip though, Most likely an indication to adjust the level of the float or perhaps even replace the kit altogether, Never hearing the tank fill up is not related to dripping.

Remove and clean the flush valve and the flush valve seat(it is the flap that opens up to allow water to flush the toilet). Once the edges of the valve and the seat are cleaned, possibly with "0000" grit bulldog steel wool, it should not leak anymore. To make sure it is not leaking, once tank has refilled completely, put some food die in the tank. If, after 10-15 minutes, the water in the toilet bowl is colored, you have a leak from the flush valve. Replacing the flush valve is very easy - just bring the old one with you to the hardware/plumbing store and ask for a replacement. Blessings and good luck!!

If you;re sure its under the toilet, look for a crack in the base, especially around the retianing bolts, or the wax bowl seal is gone, needs replacing.

Do you have a way to check your water pressure each of these problems could be caused by excessive pressure you may need to install pressure adjustment valve if you can't find another way

As long as the chain on the flapper is slack after it flushes it will seal.
then put some weight on the flapper just to experiment to ensure a tight seal.
if you can see the drain elbow under the toilet in the basement with the ceiling exposing the drain that would be the guaranteed way to know for sure. I am assuming you have done that.
Last but not least the small line on the fill valve that goes into the overflow could be weeping from the fill valve as this is the bowl wash.. So it will drip or sometimes it runs a bit . If this helps

Basically, if water is flowing out of the overflow tube, then water must be coming in...I would suggest that you check the intake valve, it is probably not shutting off completely when the float comes up to the prescribed level,and continues to leak some water into the tank. That is what the overflow tube is there for.

Anytime that you have water dripping between flush cycles, it is caused by water leaking through the flush valve. Without being able to look at it I would recommend placing the entire flush valve assembly (flush ball/plunger and the valve seat (washer). The part is not expensive, although the labor may be. You could replace the flush ball or plunger (whichever type it is) yourself and that may correct the problem.

WHAT IS DRIPPING,?THE TANK ?OR IS STILL FILLING UP ?IT SOUNDS LIKE YOU NEED TO ADJUST THE WATER LEVEL TO THE PROPER 1.6 GALLONS PER FLUSH,WHICH IS MARKED ON THE INSIDE OF YOUR TANK SOMEWHERE,CAUSE WHAT YOUR HEARING IS THE WATER RUNNING DOWN THE OVERFLOW TUBE

2 possibilities.

1) The diaphragm in the Sloan is for a urinal. (1.0gal vs. 1.6gal);

2) Not enough water. (stop partially closed, supply undersized)