You can turn the water off for that sink if you don't care if it's fixed.
I also suggest that fixing a clogged pipe in the wall does not mean having to do a major remodel. We had a clogged pipe in the wall that prevented us from getting water and so it's a different problem than yours tho the fix is the same. My cousin cut a 12" square hole in the wall, turned off the water at the main, drained the pipe, cut a chunk out and replaced the pipe with a new one. I think for yours, since it's the drain that's plugged you'd gain access to the down pipe and unplug that.
When I think about your situation I do think it would be best to call a plumber. I'm not sure that a plugged down pipe would be the cause. That pipe is large. The drain pipes outside the wall is much smaller. Nothing could be released into the down pipe that would plug it because it wouldn't get thru the initial pipes.
Did you try putting a snake thru the pipes? I also recently had a plumber come out and snake all of our drains because they were draining slow. The snake is a wire that is several feet long and would go into the pipe in the wall.
My wall is papered and so it was just a matter of not cutting the paper and then letting it back down to cover the hole. Eventually we'll patch the hole with dry wall.
Later: After thinking about this. What you've reported isn't making sense. If the pipe in the wall is plugged then the sink wouldn't drain at all unless there were a hole in the pipes. And for the water to be draining under the sink and not in the wall, there has to be a leak in the pipes under your sink. There has to be a hole/leak somewhere for water to be under your sink.
I suggest that the problem could be that you need new washers in the knobs that turn off the water under the sink. The leak could be from the incoming water pipes instead of the drain.
I also don't understand why you're using the sink if the water is leaking under the sink. Turn the water off. There are two knobs for hot and cold water under the sink and in the wall. Just turn the water off. If you can't turn it off then you know there is a problem with the pipes outside the wall.
I would just call a plumber without telling my husband. At least your husband's ire would be short lived because the problem would be resolved. By letting this go on and on you're having bad feelings over and over. One big explosion because you called a plumber and you'll have the satisfaction of having the job done.