You can get flushes at auto parts stores. Maybe Walmart, too. You pour them into the system, fill it up with water, and run the engine a while, then drain and flush, then refill with coolant/water. Follow the directions on the label.
The only concern I have is that you might remove that one bit of crud that's plugging a leak somewhere in the system.
If you want to forego using a flush additive, then I would simply fill the system with water and drive it for a while, then drain and fill with coolant, etc.
HTH.
Doug
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