I loved our Moby in the beginning until DD was about 15 pounds, then it stretched too much.  But it was great to get her in there really snug and it held her very securely so I could do stuff with both hands free.Â
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I like a ring sling for quick in and out but I don't like the one shoulder weight distribution and I can't use it at home really because I don't have two hands available for doing stuff.
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The Beco Gemini is my favorite for quick in and out and trips to the store because it is really easy to get on and very comfortable. DD is 23 pounds now and I barely feel her weight even in a front carry. I like it better than the Ergo because I am short and the Ergo is too bulky and the straps always feel like they will slip off. I've also used the Gemini for long periods of time at fairs and places like that in the summer and it didn't really get too hot. The downside is that it is about $130 but it is so versatile I really think it's worth it. You can do front facing carries with it also.
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At home when she is fussy or super clingy and I absolutely have to get stuff done like dinner, etc...I use a woven wrap or mei tai. They both give a very secure back carry so that I can do just about anything including cooking, laundry, cleaning and dancing with DD1!
I just got one that has a structured waist and wrap ties and it is awesome.
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The woven wraps are very comfortable and really get her relaxed and sometimes sleeping.
If you check the FSOT on thebabywearer.com you can get very good prices on mei tais and wraps and honestly, a used wrap is even better than new because it is broken in for you! One I got off FSOT is so incredibly soft that the first time I put it on I said wow! it feels like a marshmallow!Â
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If I really had to choose on a budget, I would go with a woven wrap a mei tai and the Gemini because you can get great deals on used ones. (not the Gemini probably because it is pretty new)
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Congratulations on your new baby!