Similar to your responsibilities to the state of Delaware, registering in another state as a foreign corporation comes with its attendant responsibilities.
Each state charges an initial cost to qualify, followed by an annual filing fee to maintain your registration. Some states also require periodic information forms for a filing fee.
Once you are registered you become responsible for the applicable annual franchise tax for that state. (This is a fee for the right to do business, separate from any income tax the company may owe in each state.)
You may need to apply for a local business license as well as any special permits and licenses for your field of business (e.g. construction, architecture, accounting, food service, medical, etc.). [Gust Launch does not handle these matters.]
Assuming you act as your own registered agent in the foreign state and do not use a third-party registered agent, you will be responsible for accepting and responding to all state correspondence and official notice at the address listed on your application.
The address you register becomes a public record and you likely will receive solicitations at that address.