the Time Machine backup does not look anything like the contents of your HD folder on the desktop
For example, this is what mine looks like:
Backups.backupdb
Peter's iMac
2018-10-24-171346
2018-10-20-110602
and on and on and on for yards!
2018-10-24 is obviously the date. 171346 is the time: 5:13 and 46 seconds PM
each one of the above lines are folders.
Inside the ones with the dates, you will find your Hard Drive icon. Inside that are all your files.

Does Time Machine seem to be running on time? Check the Time Machine pane in System Preferences where it will tell you when the last backup took place, when the next one is and the oldest one.
Take your Passport with you while you are away, assuming you are taking your laptop, and keep the backups going.

This could take some time so if you want to do it by email, which allows pictures easily, let me know and I'll make my email address available.

P