How can I make Time Machine do a full backup?
The mechanism to force a full backup isn't well documented, but it does get logged to the console logs as Forcing deep traversal on source: "Macintosh HD" ... but even if this happens, the system will base the storage of new copies of files on what exists on the backup volume previously.

I would add a new drive and back up to it once if you are concerned that your backup is no longer reliable.

From the command line, you can then check on things with tmutil compare or by using a tool such as BackupLoupe to convince yourself that specific (or ongoing) backups are correctly being made.

You can hint that a full scan is needed by excluding all the files you care about in options. Then run a backup, then remove the folders / files from the exclusion and then tigger or wait for the next backup.