The first thing you should check is for a leaking flapper.
First, make sure that the small fill tube from the fill valve is just above the overflow tube in the tank.
If the tubing is pushed too far down the overflow tube, it can siphon water into the bowl, bypassing the flapper.
If that is fine, then put food dye in the tank, and if the dye shows up in the bowl, then you need to replace the flapper.
You tank will sweat if it's constantly refilling the bowl.
Replacing the entire toilet seems kind of dramatic.