Ah, we outgrew our Moby early... earlier than I would have liked. I think I moved him to a woven wrap around 3 months. (He was almost 20 lbs at that point. He was outlandishly huge as an infant.)
My woven wraps (all homemade) were my favorite carriers by far... so comfy and I felt like they actually looked cute on me. But I couldn't get the hang of getting my gigantic kid onto my back in the wrap by myself.
I've been happy with the Ergo, though agree that being able to take it off and hand it to someone else (as with a Beco) sounds really appealing. Honestly, my biggest complaint about the Ergo is that I think it's extremely unflattering on me. Straps make me look squishy. :P We have gotten a TON of use out of the Ergo though, I find it really easy for a back carry or a front carry, and my husband'll actually use it, unlike my wraps.