The image you can download from the Ubuntu website already is a live-CD waiting to be burned, you can burn it by following the instructions which can also be found on the download page:
I'd like to avoid using a CD if possible, but if a CD is necessary to create a LiveUSB, then I expect that I would just skip the USB entirely
A physical CD is not necessary to create a LiveUSB; you just need the ISO image. Use UNetBootin to create the LiveUSB; just point it to the ISO and to the drive/device that is your USB stick.
However, depending on how old your computer is, it may not support booting from USB; in that case you can use a CD to "kick-start" the USB.
Best Answer
The image you can download from the Ubuntu website already is a live-CD waiting to be burned, you can burn it by following the instructions which can also be found on the download page:
The resulting CD will be your live-CD.