I have had to oil a walking foot because it was squeaking and really annoying. However, I only used a drop of oil in 2 places and had no problem with it leaking oil. I would try cleaning the foot thoroughly and then using it to see if that helps. I would also make a sandwich using the same batting (wadding) that you are using and having a problem with. The sewing machine store warned me against over oiling my machine.

There is website for Bernina's and if you do a search you will find it. Bernina Jeff put out a you tube that shows people how to take apart a Bernina walking foot and oil it. There are moving parts inside and eventually, the oil with thicken and may cause squeaking and even lock up the arm that moves. In the video he mentions other walking foot should be somewhat similar. He recommends doing every year or so depending on usage. He recommends if you don't want to do, to bring it to your machine tech.

You could blow the extra oil out with an air compressor (or canned air).