My vote is NO.. you should be fine to use a standard 120v 3 prong outlet at your stop in Raleigh to top off, and actually wouldn't even really need it unless you are rolling into town on "fumes". If they have a convenient plug to use then go for it, but if you are hunting for an outlet and parking sideways in the front yard to use the 120v plug then I would just skip that all together.
With a 4 prong 240v You would probably spend a good 20 minutes of hassle of getting behind the dryer at someone's house and running a cable outside and plugging in your car. Then about the same time to un-hook the cord and move the dryer back into place and wrap up all the cords. There are PLENTY of
Superchargers along your route, when leaving town and heading south you could get a little juice at
Garner, NC or
Benson, NC and make sure you have just enough to get to Florence, SC....
When traveling from Raleigh to Miami I would strongly suggest you stop at the location at
Buc-ee's in Florence, South Carolina since that gas station chain is very
unique and you could easily spend 20-30 minutes in the store while you charge up to a higher charge. Also, skip the 40 stall charger in
Dillion, South Carolina, its just a big parking lot and really isn't impressive