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Try to repeat plunging a couple more times and run hot water through. You will probably still be pulling very black water back. If not your drain could be undersized or have another more significant problem.
The splashing was a general precaution, especially if you have used chemicals to try to clear the drain. Some sinks are really shallow, and you may not have room to add water.If you do have room to add water, hot water will help loosen clogs.
I've done plumbing in the Air Force for over 14 1/2 years. This was a typical problem. Our facility managers were required to attempt to plunge any clogs before calling us to come out. Most of the time they didn't know about plugging the overflow drain, and we would still need to come out.