A good nights sleep is going to make all the difference in the world with your son's behavior. Also, keep the bedtime the SAME every night of the week. You can't change it on the weekends. Consistency is KEY!
A five year old needs between 10-12 hours a night. As crazy as it sounds, you should put him to bed at 6:30, even if he is restless, hopefully he will be asleep by 7:30 which will give him 10 hours a night. I really believe most kids are sleep deprived these days.
Children challenge their mothers. Just keep doing what you are doing. With the babies, he is acting out for attention....and, he's getting it.
Try to see if your husband can take the babies and you can spend some one on one time with your son, maybe once a week. Something mellow, like go for a walk, bike ride to the park. Get a quiet bonding connection with him and talk, NOT about his bad behavior but just try to give him some extra attention. I know that may seem impossible these days but he's acting this way for a reason.
Good luck, you sound like a great mom.