Doing the same thing to 5ver. A lot of the factory caulking is a very thin layer. If it will peel up and come off I try to get it off with a plastic puddy knife and using goo gone and/or WD-40. Then wash off with a heavy amount of Dawn dish soap to cut the citrus or oil and then wipe dry with a towel. If I have already caulked on that place I get my old caulk off first.
Otherwise I am just cleaning the factory caulk back to a bright white using above method.
I allow time to air dry and been masking tape around the area just leaving exposed the caulk. It's time consuming but it's been 10x easier and cleaner when apply and working the caulk. Use the Dawn soap water to smooth the caulk and after removing the tape to smooth it back down.
This is the factory caulk before cleaning.
Up on the ladder today.
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