Re: how to remove adhesive
In a word, you can't.
What I would recommend is to either remove the adhesive or to overlay the area with new 1/8" or 1/4" plywood and restain the area to match the existing cabinets. You can get stain matched pretty easily at any major paint dealer (not a big box retailer ). All you have to do is have a scrap of the same type of wood for them to test on and a sample of the original finish, which you can get by removing a cabinet door or possibly a piece of trim.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!