Some weeks ago I was in BC, Canada and I saw a lady walking her dog and I asked if I could take a picture with her dog. She was genuinely kind, even asking me if I wanted to get hold of a leash, and when I stood up after having taken a picture, she asked if I wanted to go down the lake with the dog. I said "(It's okay) you can go on your way" as when I first met her she was not going to the same way I was and I didn't want to take more of her time, but after we parted I wondered, was it a polite thing to say? Most cases I've seen it is used as in "none of your business, you go on your way" and obviously I didn't mean like hit the road..! I wonder if the expression has some impolite connotations, and additionally what was better way to say what I wanted to mean?
Confusing, at best. Grammatically there is nothing to choose but semantically, the difference could cause wars!
You seem to have meant something like "Don't let me keep you from (going on) your way."
At very best - not certainly - a policeman might get away with "Go on your way" to anyone, including a respected citizen.
Similarly, a thug might use "Go on your way" to mean "Clear (or worse) off/get lost…"
No, thank you, you have been more than kind.
Thank you, I have imposed upon you enough.
Your concerns are well founded. We all can become lost for the correct words in spontaneous situations. What you said at the time was not what you would have said had you been anticipating the lady's question. A lesson we all should take is not to be quick to respond.
you can go on your way
is probably not what you would have said had you thought on a response for a couple of seconds, and you seemed to understand that almost immediately. It is not correct to give instructions unless asked for them, unless you are charged with doing so by a position you hold. An exception would be emergencies when correctness can be suspended (get down!, watch out!, for example)