Thanks a million. You guys are great. I didn't have BOSS Userlist Manager set up completely.
For anyone in the future who runs into this, here's what I did to fix it:
Open BOSS Userlist Manager from inside Mod Organizer. Enter the appropriate information for all fields for Skyrim. Leave "User Game" blank. Assuming you only plan to use this for Skyrim, use the dropdown box to select any other games you have installed and delete the information that is pre-filled for those games. Save. This should take you to the main user-interface screen. Select "Unofficial Skyrim Patch.esp" from the bottom right menu. It may look like this: >IF VAR(USKP20) MOD: Unofficial Skyrim Patch.esp. I have no idea why. :) Click the "Create Override" icon at the top. Select "Update.esm" from the bottom right list and click the "Copy" button next to where "Update.esm" appears at the top. Click the "Save Rule" button. Repeat for each of the unofficial patches, creating rules to insert them after their respective DLC esp's. Make sure to click "Save Rule" each time. Exit BOSS Userlist Manager and save when prompted. Run BOSS from inside Mod Organizer. Run the Dual Sheath Redux Patch again and it should complete properly. Double click on "Overwrite" at the bottom of your mod list and drag and drop the folder found there to your "DSR Output" mod to combine them. I don't know if that's a "Best Practices" method or not, but it seemed to work for me. I just moved all of the mods to follow their respective patchees. I have no idea why each of the patchs seems to have an "If" statement for a name. I guess it's so it will check to see if you have the appropriate patchee before it installs the patch. Seems like they could have done that behind the scenes somehow. Oh well. Thanks again to everyone who helped. You guys really are awesome.