A good paint cleaner will make short work of any old lsp. Couple it with a mild to mildly aggressive (but clearcoat safe polish), and your paint should be totally lsp free when you're done working on the paint.
I tend to stay away from Dawn, as it will remove some (but not all of your wax or lsp), it also tends to dry the paint with its alkaline ingredients. A good ph balanced car wash solution will do a lot more for you when you pair it with a proven paint cleaner (such as Gliptone's pro-buff).