I got this problem solved and just want to let others know what it was.
I followed ewdel1's advice and brought my phone in to have it looked at and sure enough, was told that this is a problem others have had and it was due to a faulty integrated circuit. Luckily, I was able to take this in to be looked at when I did because my 1 year warranty was about to expire in a week.
Apple replaced my 4S with a "new" phone. The replacement looks and works good as new, so happy I didn't have to go through a hassle and the replacement was very straightforward.