Yes, you can. Either is, as often, being used symbolically in this kind of sentence. They are extremely similar in meaning, in this usage - I might almost say identical. "To sense" might use any of the five sense - sight, sound, smell, taste, touch. "To feel" uses only touch. In use, some people would say "I sense" to imply that they are using some kind of intuition or "sixth sense" to understand something. And "I felt that" might be taken to imply an emotional response, a feeling.