I don't know if the Ultimate Baby Wrap is the same as the Moby. I have seen the Wrap online and the picture look almost the same as a Moby. The Moby is stretchy, but not very elastic. I've never had to readjust it while wearing it. Although, the biggest baby I've ever carried in it is 18lb. maybe that has something to do with it?
Welcome to the site! I hope you enjoy your time on here and find a lot of encouragement! There are a lot of really neat people on this site. I know because most of the active users are related to me. :-)
I noticed in your profile picture you're wearing a sling. Have you tried a Moby? Once I tried it I couldn't believe how much more I liked the Moby than a sling.