Well, generally speaking, contact lenses can lead to light sensitivity. As we know that there are many reasons which can lead to it. And it's a symptom of many different diseases, disorders, problems and conditions. For example, an infection or inflammation that irritates the eyes can cause photophobia. Also, it can be a symptom of an underlying disease such as a viral illness or a severe headache or migraine. But you should know that contacts irritation can be the cause of the light sensitivity. For example, if you have worn contact lenses longer than recommended or in people whose eyes are particularly sensitive, lenses may develop deposits on the surface of the lenses and cause irritation. In that way, and in a long term, it can cause light sensitivity. So for your situation, a prosthetic contact lens that mimics the color of your eye can be effective. And also make sure your contacts are not irritated.
Well, it seems that there is something wrong with your eyes. As a matter of fact, I used to have the same problem with you, and I realized that it is just because I needed some more time to get used to wearing contacts. However, some people will have some allergic reactions to contacts, and thus have some side effects, such as light sensitivity. Maybe you need more time to get accustomed to them.