This was all a lot easier than I imagined it would be. For one thing, my issue of being unable to renew my DHCP lease could have been solved with dhclient -r, from a response I got on the FreeBSD-questions mailing list:
The -r flag explicitly releases the current lease, and once the lease has been released, the client exits.
So if it happens again, I can test this (though I’m happy to skip that experience).
A little sleuthing turned up a python app (3412 lines is hard to call a script) that will update my DNS records at ZoneEdit.
So buying a domain from GoDaddy: $7.95/year.
DNS from ZoneEdit: free.
Running my own domain, and being able to see if Google AdSense is worth all the hype: priceless.