Using track templates, maybe?

Another possibility is using the snapshots feature of SWS extensions. Can import/export complete project mix settings. example:

you could also just get a mix set, delete the items. Save as _blank project. import in other items, save as "song x" reload _blank, import, rinse, repeat.

Thanks guys. I ended up using templates because that was the simplest thing and I haven't an excess of time on this one. Very similar to a blank project..

Get SWS and even can grab Kenny's video on SWS extensions if you want more info on them. I couldn't use Reaper without SWS Extensions.

Think of it the opposite of your title. Copy the new song to a previous mix. For quick roughs like this, either just import (or drag & drop) the tracks into an existing session or throw all the songs into one session to start with (probably rolled them in that way anyway). I wouldn't spend time playing with features like making templates and so on. The last session that looks like a good starting point is your template.

Project Tabs? 

Would opening a Project Tab help?