Hi Jonathan,
Welcome to the Moz community!
You should definitely update the display and landing page URLs to match the landing page the user will eventually end up. Not doing so could be in violation of the Google AdWords policy and negatively impact your account.
source: https://support.google.com/adwordspolicy/answer/6021546?rd=1#027
- Redirects: You can use a redirect for tracking purposes as long as the destination URL's final resulting page uses the same domain as your display URL.
I know that in the above they talk about redirects for tracking purposes, but it remains the same for general advertising guidelines.
Updating the ads shouldn't be much of an issue, though, with Excel and the AdWords editor. I suggest exporting all of your campaign ads into Excel and then doing a Find and Replace to update all of the URLs to the new domain. Then you can easily import those ads back into the AdWords editor.
In fact, I believe the AdWords editor added this as a native function in the 11.0 version.
https://support.google.com/adwords/editor/answer/47657?hl=en