I have been asked to move the FC/tables for a survey from one SDE to another. Can I do this by moving the data, re-linking the layers/tables in the mxd then overwriting, or do I need to republish the service then republish the survey (referencing the new service)?
The survey is currently published and in use.
One you have copied your data into your new database and created a feature service to mirror your original one, I would start by creating a clone of your production survey. This clone would be just for testing, to make sure you can replicate your survey on top of the new feature service. You can do this as follows:
If you got your new clone survey to work well, then you can now go into your production service, and copy/paste the submission_url and form_id values into the production survey and publish. At that point, data submitted from the survey123 app will start going into the new feature service. End users do not need to download your production survey again, because the target feature service of a survey is discovered at runtime.
Note: If for whatever reason you are not able to smoothly make your clone test survey work against the new feature service, you may want to publish a new version of your survey and ask users to download the new version.
Before doing anything of the above, make sure you make backup copies of your data and surveys so you can revert changes in case you need to. Any changes you make anywhere in a production system (database, feature service, survey, device...) implies risk. The above is provided as a suggestion but you may encounter issues through implementation. Be ready to have a back-up and recovery plan in case things do not go as planned.