I have 3 kids who are similar ages to yours, and one on the way. We'll have 4 ages 4 and under in March.
When my second was born, he didn't sleep for 9 months straight and I felt a lot like you described - always an edge, always needed, always spread thin, always made to feel guilty. It was a tough chunk of time - and one I never thought would end. It finally did end, and I returned to feeling "normal" -- but that was after he started sleeping, and after I started reaching out for help!
I would definitely see if you can connect with other Mamas - play groups, story time, MOPS, coffee hours, etc. I know it's a lot of work to get out of the house with 3 kiddos so small, but it's worth it!!
I would also suggest some "ME time" - a night out with your girlfriends, mani/pedi/massage, going to a bookstore and curling up with a (decaf?) coffee and a favorite book. Basically anything that is "good for the soul" - and you don't have to worry about wiping noses or changing diapers or cutting up someone else's food :) Set a schedule to do this regularly. Just like you have to stay connected with your husband during early childrearing years, you also should stay connected with your girlfriends.
Talk with your husband about how you're feeling. Find a time when it's just the two of you and it's quiet (or as close to those as possible, haha!) and share your thoughts and feelings. Tell him you're not asking him to "solve your problems" (men have a tendency to want to "fix", no?) but that you just want to be heard, and understood. This might also be a good time to talk about scheduling that ME time, and get him on board!
While I don't believe medicines are the "be all, end all" to one's struggles, they certainly can help - even if they are temporary. What you're describing sounds like a chemical imbalance, resulting in anxiety and depression. I totally understand not wanting to take meds, so if that's not an option for you - can you meet with a medical professional to discuss your feelings, and point you in the right direction? Your OB, family doctor, Nurse Practitioner, a therapist. Or you can call your insurance company to find out what the coverage for a therapist. I think it's definitely worth finding someone who would be able to listen to you in an unbiased way, and help you build up some coping skills for when life gets stressful. Sometimes it's not enough to feel like you can just fix yourself...
3 young kids is a LOT of work, I totally understand Mama! Hang in there, you're doing a great job!!!
(((Hugs)))