While I dislike writing unit tests, I have to admit they are useful in protecting your butt when something breaks, since the test should catch it first. Of course you need to decide whether in a particular scenario they add value or just make you manager happy.
In a case like yours, you can make code modifications and hope nothing breaks or build unit tests and ensure that you don't break any of them when refactoring. Initially rather than just ripping out the seemingly duplicate methods, rip out/tweak their implementation and have them point to what they seems like a the right method to provide the common functionality. If your unit tests show breakage, then you know that you missed something.
If you do things wholesale, then you are likely to break something in an unmanageable way. Oh and make sure things are version controlled ;)