Yes, you can try oil pulling. I would recommend that you add some clove oil to whichever oil you choose to pull with.
Clove oil is used in dentistry to help reduce pain. In instances of dry sockets, after tooth removal, the socket is packed with gauze and clove oil is used to disinfect the socket and provide pain relief. It has a numbing effect, although it burns when first applied. You can apply a drop of clove oil directly to the tooth that aches, and then add some clove oil to the pulling oil; three drops is sufficient.
There are no negative side effects of oil pulling, unless you inhale the old oil at the end of pulling.