Reminding someone to stay on a/the “straight and narrow [path]” is similar to cautioning someone to straighten their lopsided loads, not in the sense that a lopsided load will eventually topple, but that it won’t fit on the narrow path or through the “strait” gate leading to whatever rewards might await those who live good and honest lives.
straight and narrow noun: informal the proper, honest, and moral path of behavior
[perhaps an alteration of strait and narrow,
an allusion to Matthew 7:14: 'strait is the gate, and narrow is the
way, which leadeth unto life']
(from ‘Collins English Dictionary’ via ‘The Free Dictiobary by Farlex’)
Your edit mentioning “Honesty is the best policy” (which although capturing the essential meaning of the original, is arguably a bit too general) prompted a search for other idioms containing “honesty” that, unlike "Honesty is the best policy," specifically include the notion of motion on a path toward a desired destination.
The following is the only one I found containing a direct mention of motion, but unfortunately it appears to have been concocted by Ed Shewan and Annie Lee Sloan, the authors of ‘Mastering Communication Skills’ merely to provide an example of a simile, for it is not at all familiar to me and more importantly, is found nowhere else on the internet:
Dishonesty breathing in a heart is like a bicyclist going downhill
(from ‘Google Books’)
There is the following quote (attributed by some to Mark Twain), however, that is legitimate and fairly well-known and which could be paraphrased to kind of include the notion of “staying on [the right path]” by using “stick to” or “stick with” instead of the quote’s original (or close to it) phrasing:
Speak the truth at all times, it's easier to remember.
(from ‘Here's How: An Introduction to Practical Discipleship’ by Lee Brown, via ‘Google Books’)
Paraphrasing it with “stick with” would give:
[Always] Stick with the truth, it’s easier to
(from ‘Family Matters: An Ernest "Sparky" Hemingway Mystery’ by Joel Rosenberg, via ‘Google Books’)
and paraphrased with “stick to”:
[Always] Stick to the truth, it’s easier to
(from ‘One Day Too Long: Top Secret Site 85 and the Bombing of North Vietnam’ by Timothy N. Castle, via ‘Google Books’)