New rim strip (squishy thing) and check the valve hole for sharp edges or burs. It sounds to me like your rim is cutting tiny holes in your tube at the valve stem, a semi common occurrence.
A new rim strip SHOULD cure the issue, but i would check the rim as well just in case. I've seen rims come from the factory with metal burs left over on the valve holes where they were drilled, especially cheaper entry level rims like weinmann 519s etc
Rim strips don't last forever. With repeated changing of tubes and tire they get stretched and shifted around and do their job less well. Their job is to protect the tube from any sharp edges of the rim and where the spokes are attached,the nipples can be rather sharp from the screwdriver or tool that was used when the wheel was built. (yes they could have chose a different word but it really is spoke nipples)
It is also a good idea when changing a tube to run your hand on the inside of the tire to make sure there are no thorns stuck in the rubber waiting to pop your next tube, but it sounds like you have narrowed it down to the valve holes.
Finally, when installing a new tube be sure that the valve comes through the hole straight, at a 90º angle to the rim surface. Otherwise when you inflate it and introduce pressure, the base of the valve is pressed into the valve hole of the rim which can be rather sharp.