A cradle hold will always have the chin to chest issue and having used that position with my babe when she was a newborn (before I knew better) I can confirm that it looks and feels wrong! Tummy to tummy is fine for supporting their neck, you will have the baby up really high on you, with their head up sort of on your collarbone. With their head turned to one side and the top rail of the sling up to the base of their head. There are some great tutorials on YouTube and I actually just watched a great one this morning on using a ring sling with a newborn. If you search PaxBaby on YouTube the most recent added video is that. She is using a woven ring sling which is a much more supportive fabric but it's the same idea. It shows you a trick to do with the excess fabric where you roll it up and tuck it under the top rail of fabric, providing extra support for baby's neck - so that could be exactly what you're after!
I had a breeze baby when DD was about 4-5 months old and while it was great in the water and did the job, I didn't find it as comfortable or supportive as other ring slings I've used. But I am only in Sydney, so the heat is not a huge concern for me.