I have been living overseas for a few years. How do I keep my US driver's license without a US address since I won't be able to prove residency in any state?
It appears that, provided you return for at least one night every five years and maintain a private mailbox in the state, South Dakota will happily consider you a resident and issue you a driver license. A number of companies, such as America's Mailbox, offer combined hotel and PMB services for exactly this purpose. These services are commonly used by people who live in their RVs or semi trucks and travel continuously, but may also be useful to you, provided you're willing to return to the US on occasion.
I am not aware of any state that allows non residents to hold a drivers license. While you may be able to buy and hire/rent cars with a license from a state you are no longer a resident in, it is not a good idea. Getting insurance might be more difficult, and I think any major insurance claim would result in your policy being retroactively canceled and leaving you personally responsible for not alerting them to the change in your licensed status.
Your best bet is to get a license from wherever you are a resident.