My son is 6 and half weeks old and started developing this non-itchy rash all over - on his face, thighs, back of his shoulders and under his diaper. It's not quite as bad as you described above - skin is not weeping and no crusty/ flaky bumps. He just has these tiny slightly reddish bumps all over and his baby soft skin is not so soft any more :(
I took him to the pediatrician yesterday and she said it's a mild case of eczema and asked me to start using Eucerin cream. Also advised Dreft laundry detergent. A friend of mine had also recommended "California Baby" products instead of johnson's and aveeno so I plan to start using baby wash, etc by California Baby. Hope this helps and hope his skin gets better soon.