The phrase "continue on your way" is correct and usable in written English.
You can use it when you want to encourage someone to keep going in their pursuits, typically after giving them advice or guidance. For example: "It sounds like you have made up your mind. Just remember to take things one step at a time and continue on your way."
Try to follow the road and you will continue on your way.
If what it's telling you saps your confidence, then ask it to step aside and continue on your way.
When you spark up a connection with a stranger on a train, do you get off to spend the day with him in Vienna, or continue on your way?
Continue on your way to the mall.
If it's actually the same thing, slap your vendors on the wrist and continue on your way.
It's the annual tradition for web developers to secure their address on our public Internet by filling out a bunch of paperwork, such as giving contact information for the registrant, administrator, technical contact, and billing contact (even the DMV doesn't require four copies of everything!) Pay the tax, er, registration fee, and continue on your way.
But as he came closer, you would have noticed his European clothes, his stylish haircut; and you might have asked yourself, "What's wrong with that man?" Had you stepped to the side as he hurtled past, tightened your scarf securely around your neck, and continued on your way, you might next have encountered a young woman on a street corner, distraught and coatless.
You looked right into my bloodshot raccoon eyes, and put your hand on the back pocket of your Wrangler jeans before continuing on your way.
This is the idea that you can separate out some of whatever you are making and make a simpler version for picky eaters, then continue on your merry way and gussy up the rest of the dish with gutsier ingredients, herbs, seasonings, etc. to give it more oomph for the grownups and adventurous eaters.
Otherwise, tap the No and continue on your merry way doing whatever else you need to do.
