Do fix it.

Don't scratch out every entry. Instead make what accountants would call a "journal entry"...

Add up all the incorrectly logged landings, and make a list of dates

Make an logbook entry with today's date, and a negative entry for landings to subtract out the incorrect number from your total time. Also list all of the dates, which I understand could make it a long entry. I would do that in your paper logbook. With that done you could then fix each date in your electronic book so it will look clean.

The fact that you fixed the error, in meticulous detail, will weigh favorably ten times more than the fact that you made such an error as a new CFI. Not something to worry about, just make sure you do the math and dates correctly.

Also you can log landings with students if you demonstrate a landing with the student not on the controls. If you know a flight was a discovery flight or first flight with a new student, it might be OK to log one occasionally.