I started the game, and entered the serial number listed on my account, but the game says that is the incorrect number.

Disregard. Twelfth time's the charm.

Don't copy-paste the key, but type it in manually. Otherwise you often copy invisible characters, resulting in pasting an invalid key. The cause could be, that the key can't be saved after you've entered it. This has usually to do with missing write permission of the actual user, whether to a file or the registry. Try running the game as administrator once or give your current user account full read-write access to the game's directory and subfolders.

Hmm, the problem doesn't happen when I disconnect from the internet. Why would that be? Makes me think it wouldn't be a permissions problem.

Interesting, I don't know whether the program validates the key online or not? Plain guessing I'd say, if it can't, it maybe sets a flag to retry next time and you've to enter the key again. That would point to a networking / firewall issue. It could be possible, that the program doesn't start this check, if it detects no Internet connection at all. Hence, no key entering when disconnected. Have seen this behavior in other programs before.
Could still be a permissions problem, when the validation result can't be saved.