I agree with Thom on causes. It's quite possible that there is a problem with your charger, battery and/or cable (did the replacement phone come with a replacement charger?)
You could also get an extra battery and an external battery charger and swap batteries. Here is a Motorola charger that works with the Bionic battery: Amazon.com: Motorola Battery Only Charger for BF, HF, BW, and HW Series Batteries - Bulk Packaging - Black: Cell Phones & Accessories
I also agree with Thom on the Razr Maxx HD, especially if battery life is important. Since being purchased by Google, Motorola seems to be more focused on lighter Android customizations and relatively quick version upgrades (though it's hard to say if that will continue with the next version update; these are always dice rolls, if it's important to you, even for the Nexus phones.) I'd also consider the Samsung Galaxy S3, which has a replaceable battery pack. I hear that the battery life on the new HTC Droid DNA is very low if you use it for any sort of media playing or heavy internet use. The Galaxy Note 2 looks fine if you don't mind the large size.