Yes, Liquid Nails and other construction adhesive is slow curing and will remain soft for days. It will work but cures slowly. Now that you have it on there, you will either have to remove it if using another adhesive or brace the panel against the island until it cures somewhat.
Titebond III is about the strongest wood glue out there. It cures in less than an hour. Contact cement on both surfaces will adhere instantly, just have to be careful about placement.
Good luck with your project.