I can imagine the frustration you feel with your wife, yes she has just had a baby and the job of caring for two little ones is draining, and I think, as I am a mother to sometimes you feel that you are caring all day for others and you just want to be left alone to relax, but her withdrawal of even affection is soul destroying, and of course you feel rejected and unloved, thats a normal feeling, anyone would feel the same way.
You guys are young and should be enjoying a healthy active sex life , babies and all, if my husband and I wanted sex I would just put the baby in the cot, safe and sound and enjoy ourselves ,if the baby started crying oh well.... they can wait a few minutes hey, sexual release makes you feel so much better and relaxed and its good for your mental and physical well being, so I made time for it , I didn't just do it for my husband I did it for myself.
Maybe you could get someone to mind the kids, go out for dinner and relax and talk, sometimes the day to day problems get in the way and we forget about ourselves and our relationship and just to be the two of you, to re connect as a couple, see how that goes, she might feel better to , some time out.
I'm sure if you went outside your marriage for sex, yes you would get that pleasure but at what cost.... is it going to be worth it ? but thats up to you and how long you are willing to wait, I can understand if you did though we all have our breaking points.
Its also the human connection we miss, the touch , the caress, the feeling of being wanted and desired all completely natural, the end result is just a bonus.
I wish you good luck, don't blame yourself though, I'm sure she still loves you but has just forgotten how to get that feeling back ,so give her a few reminders .