Is Meditation a cure or just fighting of symptoms? I meditate daily for 30 minutes, and right after each session my mind is very calm, centered, stable; I just exist in the present moment without much thinking.
However, after a while, the compulsive thinking comes back, wants to drag me towards unhappiness. I feel this is like an illness. Therefore, is meditation a cure to this illness (will it go away one day if I meditate more/longer? What should I do?) or is it just a temporary easing of symptoms? I want to cure the illness. Thanks.
That depends on the type of meditation you do. To most people, meditation is all about focusing the mind on one thing and enjoying the bliss that comes with it. This type of meditation is called Samatha(tranquility) meditation. There are many forms of Samatha meditation. Ex: breath meditation, loving kindness meditation, repeatedly chanting a word etc. 

Samatha meditation is taught in Buddhism too, but the meditation type that is unique to Buddhism is Vipassana(insight) meditation. Samatha meditation is mostly about fighting off the symptoms and looking for a temporary relief. Vipassana meditation is about curing the sickness that causes the symptoms. Here's a handbook on how to practice Vipassana.