Should I be feeding my baby whenever he needs it even if my breasts don't feel full?

Yes. For healthy babies with no feeding issues, then yes feed on demand. Your breasts will adjust in three days to his needs and you will get full. If your baby is happy and growing, then he is getting what he needs. If he is nursing literally around the clock and you never feel full, this can be stressful, and i would want you to see a lactation consultant to assess latch and efficiency of feeding.
Yes. The best infant feeding schedule is an "on demand" schedule, meaning whenever he or she wants. Don't worry that your breasts don't feel full and remember that the act of suckling will stimulate milk production. If you are concerned about your supply, check with your pediatrician and have your baby weighed. Lactation consultants can also be very helpful.