Not suppost to bubble.
Got a picture taken looking down into the tank?
If there's water leaking from the tank to the bowl you need a new flapper.
If the float is stick and causing water to not shut off and leaking by you need a new fluid master ball cock.
Less then a $20.00 fix.
It is almost impossible for the toilet bowl to flush normally but overflow due to a slow leak through the tank flapper. You may need to have the bowl snaked to be sure there are not some obstruction (possibly many feet further down the drain pipe) that settles in place but is pushed out of the way with a fast inrush into the bowl during a normal flush. Also the venting for the drain pipe system needs to be checked for obstructions such as a birds nest or dry leaves.
Don't let the weatherman string you along. Stick to your lawn watering schedule until it actually starts pouring.