For Testing:

As david.benson@appsscripttesting.com use (see Tools, Script Editor):
- TBA -

For Deployment:

As david@jgraph.com use (see Tools, Script Editor):
- TBA -

Open the web app script in the script editor.
Make the changes you wanted. Test the code to ensure it functions as intended and is bug-free.
Click File > Manage Versions. Enter a new version description and click Save New Version. Click OK to close the dialog.
Click Publish > Deploy as Docs add-on. Update the Project version to the new version you just created.
Click Update.