If I sign on with one of the less expensive droplets (the $5 / month plan), would I be able to upgrade that to the $20 / month plan without spinning a new droplet and migrating all of my content to the new one?
You have two options. You can use the "fast resize" option to quickly upgrade your droplet to have more RAM and CPU cores. This can generally be done with less than a minute of down time. The draw back is that it doesn't make any changes to your disk size. You can do this using the "Resize" option right in the control panel:
If you need to also increase the disk size, then you can use a snapshot like JT suggested. Just check out this tutorial: